They did. In 63 CE the emperor Nero and shahanshah Vologases after an indecisive war in Armenia hammered out an agreement by which the Parthians would select the king of Armenia and the Romans would crown him. This was an acceptable solution to all, as it maintained the Roman fiction of universal monarchy while tacitly acknowledging the closer relations between Armenia and Persia. Nero had correctly recognized that the earlier settlement, in which Rome had a much more direct role in appointing Armenia's ruler, was both pointless to Rome's true interests (ie, maintaining the fiction of universal monarchy) and detrimental to the eastern border, as Armenia had real practical importance for Parthia and, all else being equal, the Armenians rather liked their Iranian brethren more.

This arrangement worked fine for half a century until the reign of the aggressively expansionist emperor Trajan. technically, the agreement was broken by the Parthians, but I hold with those who think they did so because they saw the writing on the wall and hoped by shoring up Armenia they could force a repeat of Nero's war. That did not happen, but the rather anticlimactic war was followed by Roman mismanagement of the peace, followed by revolt in Assyria and a broader Jewish revolt. The momentum of Roman conquest was then utterly ended by Trajan's death.

In short, Nero and Vologases had hashed out a perfectly acceptable agreement that was ended through Trajan's aggression.

Well, the Eastern Roman Empire did beat the Persians. Emperor Herakleios (610-642 AD) led a campaign against Khosroes II for most of his reign and managed to bring about their surrender and defeat by 627 AD. It's only because of the (first) Arab Spring that we don't hear much about it. The Sassanid/Sassanian Persians that survived Herakleios' invasion were quickly swallowed up into the new Caliphate, and much of the Greek/Roman Levant and Asia Minor followed soon after.

Source: Theophanes, Anni Mundi, 6095-6305

Edit: changed 'destroy' to 'beat' after to u/shlin28's correction.

There aren't any in depth asnwers right now so I will give a short TL;DR. The actual answer is far more complex and is probably worthy of many a thesis and such.

Essentially both empires were super powers, and while they fought each other often they usually had much more to gain from peaceful relations that conflict. They also had other frontiers with less "civilized" folk, and so those borders were a lot less stable and required more attention. It was in their interest for a stable mutual border for trade, not outright conflict. Most people who tried conquest were glory hungry, like Crassus. After a while most of the leaders of the empires figured out peace would work out in their interest and for a long period of time all of the fighting between the two states was boiled down in proxy wars essentially in Armenia and other buffer states in the region. Full scale war for minor disagreements was too costly and both states developped complex, often friendly relationships. One example is that one particular Sassinid king asked the roman emperor to foster and adopt his heir to protect him from palace intrigue. It sparked a diplomatic incident when it became apparent that this boy, since the roman emperor was heirless, stood to inherit both empires.

This subject is infinitely more complex than I made it out to be, especially considering that almost all of our sources are roman. What the persians may have written, has almost totally been lost to time. I welcome someone who is more versed in the subject to expand on what I wrote, and possibly see if they can answer a very specific question I have that my prof couldn't.

I know a bit about wars from the sixth century onwards, so I'll focus my answer on this period, though it might not apply to earlier eras!

Aside from two brief periods of contention, the two empires were at peace in the fifth century, mostly because they had other threats to deal with - the Romans had to deal with internal conflict and the dissolution of the Western Roman Empire, whilst the Sassanids dealt with the Hephthalite Huns, which intervened regularly in Persian politics, killing one Shah and imposing another, Kavadh, on the throne twice. They played a critical role in the next period of conflict, as it was provoked in 502 by Kavadh to get the fund necessary to pay off his Hephthalite sponsors. Many cities did indeed pay off the Persians rather than fight them to the bitter end, as the Roman response was rather weak. Later, there was a period of stalemate once the Romans got their act together, so peace was made. There was no need for further warfare, as Kavadh got the money he wanted from the Romans.

Kavadh re-opened the war in 530 for several reasons - allegedly, the Emperor Justin I refused to adopt Kavadh's son (in order to ensure that he will actually succeed his father), whilst tensions were also high between their many client states, as the king of the Iberians had converted to Christianity following the example of Lazica, both kingdoms in the strategically important Caucasus mountains. Justinian had also resorted to building forts on the frontiers, which was prohibited, so that must have annoyed the Persians too. Hostilities officially began in 530 and the fighting was on a much bigger scale, with proxies such as Huns in the north and Ethiopians in the south (who invaded the Arabian Peninsula) playing important roles in tying down the other empires' clients. Again however, local grievances were at the heart of the conflict - both sides wanted to make sure their interests were protected on their borders. To destroy the other empire was unrealistic and probably not even considered. Like before, there were major Roman victories (Dara in 530) and defeats (Callinicum the following year), though peace was hastened because Kavadh died in 531, which placed his son Khusro I on the throne. Khusro had his brothers to deal with and was an unproven ruler, so a treaty was quickly signed - the Eternal Peace of 532. Cash payments were made to the Persians and status quo restored, suggesting that the Persian situation wasn't that weak after all. The Romans were probably not strong enough to push any further into Persia anyway.

Khusro I launched a sudden invasion of Roman territories in 540, resulting in the Sack of Antioch, an act that sent shockwaves across the empire. There were many reasons for this, including entreaties from Goths in Italy, who were facing defeat at the hand of Justinian and sent an embassy to the east to ask for Persian help. More likely though it was due to general Roman weakness in the area, as Justinian devoted his efforts to his western reconquests. Khusro proved to be an able ruler and centralised the Sassanian administration. According to one of my tutors, it was also a tradition for Persian Shahs to wage military campaigns to demonstrate their strength, so after defeating his internal enemies, Khusro saw a chance to strike at the Roman Near East to demonstrate just that. After his initial successes though, the conflict became a localised war in the north rather than in Syria/Mesopotamia, where truces were repeatedly signed. By 562, both sides were ready for peace, but again the terms were close to the status quo - the Lazis were placed in the Roman sphere of influence again, and both empires' clients were restricted from further warfare and trade. It was an inconclusive end to the war, but at least both sides were satisfied - Justinian was a tired old man by this stage, whilst Khusro was a highly effective ruler and I think he recognised that fighting over Lazica was pointless when he could achieve his goals in peace as well - the Lazis had switched loyalties repeatedly over the past few decades, so there was no reason why they wouldn't do so again (and they did).

Then came Justin II, who decided to back Armenian rebels (against Persia) out of Christian solidarity in 572, provoking a war that lasted until 591. Fighting was mostly restricted to Armenia/the Caucasus, so only a few major gains were made on either side. Stalemate however was broken when a Persian general overthrew the rightful Shah Khusro II, who fled to Emperor Maurice for protection. Khusro II was restored with Maurice's help, and in return Maurice received the bulk of Armenia and the restoration of important border forts. Then followed a period of co-operation, since Maurice and Khusro were on friendly terms with each other, demonstrating how useful co-operation can be for both empires, particularly in oppressing those pesky Armenians.

Maurice's execution by the the usurper Phocas in 602 however changed things, as it provided an excuse for the ambitious Khusro to expand his empire. The course of the war is described in my comment here, but I have to emphasise how this war changed everything. Unlike before, Khusro made massive territorial gains and perhaps even thought it possible to fully consolidate them into his empire. It became realistic for the Persians to actually destroy the Roman Empire, since the Romans had lost its richest provinces. Allegedly, Khusro rejected several generous peace overtures from the Romans, including one that effectively made it a vassal-state of the Persians. However, this was not to be, thanks to Heraclius' miraculous victories in 626-8, but it was probably the only war between the Romans and the Sasanians that even had the possibility of ending one empire. The cost was however ruinously high, as both empires exhausted themselves and left them open for the armies of Islam.

So why were the Romans and Persians unwilling to destroy each other? I think the simple answer is that it just wasn't possible. From our sources, both sides recognised each other' status as one of the premier empires of the world (though obviously they thought of themselves as the best and most important). Their embassy system was surprisingly complex and indicated that if there was a dispute, there was a recognised system to deal with it. Both empires also had plenty of other issues to deal with, so often it was easier to get their clients to do the fighting for them, which in turn prioritised the clients' need - Arab raids was an often cited cause for conflict, since they continued to raid each other even whilst their overlords were trying to make peace. Great victories helped to assert superiority, such as under Maurice and Heraclius, but it was never one-sided enough to dismantle the other empire. Instead, these emperors focused on forging settlements the Romans can live with, ones that will give them the strategic advantage in case war breaks out again.

I don't think I answered the question fully, so feel free to ask any follow-up questions that you have :)

Yes, the capital was in Ctesiphon. But that was not the center of the Sasanian and Parthian states, which reached eastward to the Oxus basin. Which along with Qadtard's comment answers a great deal of your question-it would have been absurdly difficult and costly for Rome to conquer and hold as large a landmass as the Sasanian Empire, and ditto for the Sasanian Empire. For instance, I'm not sure that any of the major Sasanian cities on the Iranian plateau like Firuzabad, Bishapur, or Ishtakar, were serious candidates for Roman attack.