Aaah, the Roman First Century. Truly a time of glorious entropy. The thing is, it wasn't just chaos for the Romans, who were, despite their disorder, still one of the most organized states in their area at the time. I'll discuss Parthia here, as it's the closest approximation to hostile Persians that the Romans really dealt with in this period (after the Seleucids had been essentially crushed in the Second Century). Wikipedia has a super handy map. The hostilities that most people are familiar with here is the brutal devastation of Crassus and his legions, an event which threw the power scales amok in Rome. It would seem that Parthia ought to have a perfect opportunity here to attack or invade Rome, so why didn't they?

Well, the answer mostly revolves around what I started with. The Parthians, despite being able to usher in some pretty decisive defeats of Roman forces (see Marcus Antonius' failed invasion for extra context), were pretty disorganized in comparison. I'll give you Cassius Dio's description of them (40.14-15, using this translation) here:

These people dwell beyond the Tigris, for the most part in forts and garrisons, but also in a few cities, among them Ctesiphon, in which they have a royal residence. Their race was in existence among the ancient barbarians and they had this same name even under the Persian kingdom; but at that time they inhabited only a small portion of the country and had acquired no dominion beyond their own borders. But when the Persian rule had been overthrown and that of the Macedonians was at its height, and when the successors of Alexander had quarrelled with one another, cutting off separate portions for themselves and setting up individual monarchies, the Parthians then first attained prominence under a certain Arsaces, from whom their succeeding rulers received the title of Arsacidae. By good fortune they acquired all the neighbouring territory, occupied Mesopotamia by means of satrapies, and finally advanced to so great glory and power as to wage war even against the Romans at that time, and ever afterward down to the present day to be considered a match for them. They are really formidable in warfare, but nevertheless they have a reputation greater than their achievements, because, in spite of their not having gained anything from the Romans, and having, besides, given up certain portions of their own domain, they have not yet been enslaved, but even to this day hold their own in the wars they wage against us, whenever they become involved in them.

Now about their race and their country and their peculiar customs many have written, and I have no intention of describing them. But I will describe their equipment of arms and their method of warfare; for the examination of these details properly concerns the present narrative, since it has come to a point where this knowledge is needed. The Parthians make no use of a shield, but their forces consist of mounted archers and pikesmen, mostly in full armour. Their infantry is small, made up of the weaker men; but even these are all archers. They practise from boyhood, and the climate and the land combine to aid both horsemanship and archery. The land, being for the most part level, is excellent for raising horses and very suitable for riding about on horse-back; at any rate, even in war they lead about whole droves of horses, so that they can use different ones at different times, can ride up suddenly from a distance and also retire to a distance speedily; and the atmosphere there, which is very dry and does not contain the least moisture, keeps their bowstrings tense, except in the dead of winter. For that reason they make no campaigns anywhere during that season but the rest of the year they are almost invincible in their own country and in any that has similar characteristics. For by long experience they can endure the sun's heat, which is very scorching, and they have discovered many remedies for the dearth of drinking-water and the difficulty of securing it, so that for this reason also they can  easily repel the invaders of their land. Outside of this district beyond the Euphrates they have once or twice gained some success in pitched battles and in sudden incursions, but they cannot wage an offensive war with any nation continuously and without pause, both because they encounter an entirely different condition of land and sky and because they do not lay in supplies of food or pay. Such is the Parthian state.

I bolded that last bit for some emphasis, in case the wall of text is too long. He continues with the aftermath of Cannae (40.28-29):

For the barbarians would not keep away from it, but made another campaign with a larger band, nominally under the leadership of Pacorus, the son of Orodes, though actually under that of Osaces, since the other was just a child. They came as far as Antioch, subduing the whole country before them. 4 And they had hopes also of subjugating what remained, since the Romans were not at hand with a force fit to cope with them, and the districts were fretting under Roman rule and were ready to turn to the invaders, as to neighbours and people of kindred ways. But when they failed to take Antioch, since Cassius effectively repulsed them and they were unable to carry on a siege, they turned to Antigonea. And since the neighbourhood of this city was overgrown with timber, and they did not dare, nay were not even able to penetrate this with cavalry, they formed a plan to cut down the trees and lay bare the whole place, so that they might approach the town with confidence and safety. But finding themselves unable to do this, because the task was a great one and their time was spent in vain, while Cassius harassed those of them who scattered abroad, they retired with the intention of proceeding against some other place. 3 Meanwhile Cassius set an ambush on the road along which they were to depart, and confronting them there with a few men, he induced them to pursue, and then surrounding them, killed a number, including  Osaces. Upon the latter's death Pacorus abandoned all Syria and never invaded it again.

After this, the Parthian Empire was generally beset by infighting (often spurred on by the Romans, who were perfectly happy to wage proxy wars) as well as fighting off migrating tribes. On top of all that, thanks to his success at Cannae, Orodes, king of Parthia, killed off the victorious general (Surena) in the fear that Surena's victory would threaten Orodes' position as king. Unfortunately, killing off your generals is one of the best ways to not have any good generals left. This on and off infighting, again, very much encouraged by the Romans, was the hallmark of Parthia's existence, with Rome often sponsoring puppet kings and/or revolts. The Parthian Empire, while not being worth the invasion, was best kept as a border province as it was too weak to actually pose a real threat to the Roman Empire.

So what were they doing besides being chaotic? Well, often being employed for mercenary work in Rome's civil wars, actually. The whole "enemy" thing gets a bit murky when money is involved. For example (Again from Cassius Dio - forgive me for relying so heavily on him, but he's one of the few sources we have that discusses the Parthians in detail. We've got a bit on ethnography and trade through Parthia, but with regards to this bit of time...it's a bit scattered):

This officer came upon Labienus before his coming and terrified him by the suddenness of his approach and by his legions; for Labienus was without his Parthians and had with him only the soldiers from the neighbourhood. Ventidius found he would not even risk a conflict with him and so thrust him forthwith out of that country and pursued him into Syria, taking the lightest part of his army with him. He overtook him near the Taurus range and allowed him to proceed no farther, but they encamped there for several days and made no move, for Labienus was awaiting the Parthians and Ventidius his heavy-armed troops. These reinforcements, however, arrived during the same days on both sides, and though Ventidius through fear of the barbarian cavalry remained on the high ground, where he was encamped, the Parthians, because of their numbers and because they had been victorious once before, despised their opponents and rode up to the hill at dawn, without even waiting to join forces with Labienus; and when nobody came out to meet them, they actually charged straight up the incline. When they were at length on the slope, the Romans rushed down upon them and easily hurled them down-hill. Many of the Parthians were killed in hand-to‑hand conflict, but still more caused disaster to one another in the retreat, as some had already turned to flight and others were still coming up; and the survivors fled, not to Labienus, but into Cilicia.

So your TL;DR: Infighting, dealing with waves of mass migration, failing to invade Syria, mercenarying out to Rome, and facilitating trade between Rome and the Far East (India/China). If you'd like to know more about that last, let me know!