Preface: I don't know anything about China so I'm not going to factor it into my post. There are also larger numbers for battles in various Greek sources but I'm no expert on those and I don't really believe most of them so I'll just talk about Rome.

There are lots of possible to answers, as huge numbers are often thrown around in the ancient sources. The last Roman-Persian war wouldn't make the list in terms of numbers, though it would be right up there in strategic and historical importance, especially considering its role in weakening the world's two superpowers just as Islam came on the scene.

In the Roman context, the largest pitched battle in antiquity in terms of casualties where we can be pretty confident of numbers might have been Battle of Cannae, with probably about 130,000 troops on the field and something like 80,000 Roman casualties, not to mention the thousands of Carthaginian casualties.* But the Battle of Arausio might be a better candidate, where the Romans appear to have suffered even greater casualties than at Cannae and the Germans probably suffered significantly more casualties than did Hannibal's troops at Cannae. At Arausio total battlefield deaths could easily be in the area of 100,000.** If we're going to believe the figures given in the ancient sources (aside from crazy outliers like many Persian numbers for Marathon or Thermopylae) then Roman victories such as Aquae Sextiae or Alesia are even greater in terms of casualties. Maybe I'm a bit of a sicko but I find the Roman defeats more interesting: overwhelming Roman victories are a dime a dozen.

The biggest war the Romans ever fought in terms of numbers was actually probably the long period of civil war that saw Augustus become the princeps, at the climax of which the Empire had 1.5 million troops in the field (Howarth 42). This was almost certainly the most troops any power of the ancient world had ever been able to field, and was without doubt never equaled. Whatever numbers you might read anywhere else, no other state but Rome in the entirety of antiquity had the logistical capacity to field such numbers.

*Daly discusses these numbers on pp. 25-32, 202. Roman casualties might have been only around 50,000 (not including those captured) so maybe Arausio is a better example.

**Compare that to one of the bloodiest days of WWI, the oft-referenced "Black Day of the British Army", the first day of the Battle of the Somme, where the British took half as many casualties, about 20,000 killed, over a front of several dozen kilometers.

Sources:

Gregory Daly, Cannae: The Experience of Battle, London: Routledge, 2002.

Howarth, Randall S. War and Warfare in Ancient Rome, from The Oxford Handbook of Warfare in the Classical World, Eds. Brian Campbell and Lawrence A. Tritle, Oxford University Press, New York, 2013, 29-45.

In contrast to /u/PapiriusCursor, I will argue that the war of 603-628 (the war began a bit later) really was as significant as it looked, a war between two imperial powers that overshadowed anything that came before it, though obviously with the caveat that I know very little about what happened before 500.

The geopolitical implications of the war speak for themselves - war was unleashed not just on the borderlands of the two empires, but also in their very hearts. Constantinople was besieged for the first time ever by an army led by the Avar khagan in 626, whilst a Persian army looked on across the sea waiting for an opportunity to make the crossing and put an end to the ailing Roman empire. Merely two years later, Ctesiphon itself was under threat by a Roman army and the great Khusro II was overthrown by his son, an anticlimatic end to a career of unprecedented Persian military success. Powers beyond Rome and Persia were drawn in as well: the Avars with their Slavic confederates, the fractious Christian princes of the Transcaucasian mountains, and most decisively of all, the superpower of the Eurasian steppes, the Gokturk Khaganate. It was not a war between two powers of equal strength as before, but one fought by the Persians in order to destroy the Roman empire - a goal that had seemed so realistic in the 610s, and which the Romans themselves knew very well, for otherwise they would not have offered to become a vassal state of the Persians in 615, the lowest point the empire had ever sunk to.

But despite all that (or perhaps because of all the destruction caused), we still lack a good understanding of the war. The sources are poor and the best of the lot are very problematic. Persian accounts of the war can only be recovered from later Arab compilations, whilst contemporary Roman accounts leave much to be desired as well: the Easter Chronicle was far more interested in calculating the date and in events within Constantinople, saying only a little about the broader military situation, whilst the poetry of George of Pisidia is literally imperial propaganda written to glorify Heraclius, so it is hardly the sort of critical account of the war that historians want. Later sources often fill in the gaps, but it is difficult, if not impossible, to know how accurately they used their sources and how their agendas affected their telling of the earlier war.

The largest problem here is that we simply don't have the sources necessary to provide the numbers, yet the two opposing sides must have mustered huge armies in order to achieve their goals. For example, we know virtually nothing about the conquest of Egypt, the breadbasket of the Roman empire, apart from the fact that it happened and that the patriarch of Alexandria fled the province in dodgy circumstances. Yet presumably Roman forces must have given battle (and lost), whilst the Persians must have left behind a substantial garrison to integrate Egypt into their new empire. The same must be true for the Levant and Armenia. It is a shame that we cannot write a more substantial account than that, but surely this enterprise was a significant one, so when we think about the numbers involved we must take this into account.

The same was true for the Romans, since decently-seized garrisons were maintained throughout the empire. Between 603-608, significant forces were placed alongside the Danube frontier (no numbers are known, but one source recorded that Phocas sent back the Danubian troops after his seizure of power, so presumably the previously highly militarised imperial presence was maintained) and in North Africa, from which Heraclius would launch his coup. One rebellious army was sent into Egypt, provoking Phocas to send an army there rather than to defend the east, whilst the African navy sailed for the imperial capital itself. Both prongs of the rebel assault succeeded, so we can only assume that both forces were perceived as at least equal to the challenge of seizing both Constantinople and Egypt. There was a revolt in Italy c.615-7 as well, which had to be crushed by local imperial forces, whilst the Lombard threat must have ensured a heightened military presence even if this had not occurred. Spania, that strip of imperial territory Heraclius still had in Spain, was meanwhile finally lost in 624, taken by force by the Visigothic King Sisebut, which implies that there was a military struggle of some sort there too.

We do not know the numbers involved in any of these places, but it is important to remember that whilst the empire was struggling for its life, a good chunk of its army was elsewhere fighting their own battles. This made the Persian war all the more threatening and makes it even clearer that the war was greater than what the sources say on the surface. I find it difficult to think of a time when the empire was in such dire straits, in which it was embroiled in conflict on multiple fronts, though perhaps a classicist can help me out here.

We do of course have some figures for the number of soldiers involved, which I think are still fairly impressive. For the siege of Constantinople, there were supposedly 80,000 besiegers, whilst the defenders had 12,000 cavalry and an unknown number of infantry. Since there must have been more infantry than cavalry (especially in a siege), this battle must have easily involved more than 100,000 fighting men. We don't have any estimates for how big the Persian army at Chalcedon waiting to cross the straits was, but we can only assume that it was reasonably strong if it is to have any hope of surviving the march across Asia Minor and to help out in the siege. Altogether, the size of the forces involved actually compares quite favourably to the Battle of Cannae mentioned by /u/PapiriusCursor. These numbers can of course be criticised and Avar/Persian numbers are only Roman estimates, whilst the suggestion that more Roman infantry were involved is only a suggestion, so we can hardly be certain. But that is the nature of history and even if all these numbers are off by a few tens of thousands, the armies involved were still sizable ones.

The most well-recorded front was in the Transcaucasian mountains, in which Roman sources provide a variety of figures. Heraclius in 624 for example allegedly faced a Persian army of 40,000 men; he defeated this army and a multitude of other attacks, resulting in him having 50,000 captives by winter. This is I think fairly unreliable, since this is only preserved in the ninth-century Chronicle of Theophanes Confessor. He probably had access to official records, but even if he did, how are we to judge whether these figures are correct? These are Roman estimates at best and only the size of one Persian army in the region was provided, who knows how many Persian soldiers were available there as a whole. Closer to the time, Pseudo-Sebeos, writing around 660, noted that 120,000 men marched with Heraclius on the campaign of 624, which is obviously a ludicrously large figure and should not be taken seriously, so this isn't particularly helpful either. For the campaign of 625, pseudo-Sebeos was a bit more sane and recorded a Persian pincer movement with one army of 30,000 men and another army of unknown size; this attack was repulsed. Later, a surprise Persian attack was also defeated by an elite Roman detachment of 20,000 men, but that's all we know.

In 626, a Persian army of about 50,000 in the Caucasus was recorded by Theophanes, which I think is significant when it took place at the same as the thrust towards to Constantinople, so (guessing wildly here) the Persians were perhaps able to muster up 100,000 men in one year, if we assume both armies to be of the same size. This is after the victories and defeats of the previous two decades, so this number is to me quite impressive. Near the end of the war, the Gokturks intervened in the war, allegedly sending 40,000 men to help Heraclius with his final campaign against the Persians, as well as presumably doing great damage on the Persians' eastern frontier. The numbers are obviously problematic, but from a strategic perspective it should be obvious that the Persians now not only had to defend their new provinces around the Mediterranean, but also the previously 'safe' territories in the east, just as the Romans still had to maintain a presence in the western Mediterranean.

This answer is a bit all over the place, but I hope I have demonstrated the problems with our sources and how the numbers we have are not illustrative of just how damaging the war was, simply because the 'true' number is unknown and must be much larger. I don't know what numbers historians have suggested for the Second Punic War, but I fail to see how a war fought on a far grander scale could have involved less people. Sources-wise, a fine summary of the war can be found in Peter Sarris' Empires of Faith (2011) in the relevant chapter, but keep an eye out in the future for James Howard-Johnston's magnum opus, a monograph focusing in particular on this 'last great war of antiquity', which I believe will be the definitive study of this conflict for a very long time.