I know that after Trajan died, Hadrian pretty quickly gave up at any attempt to control the recent Roman conquests in the Middle East but what happened with these provinces? Were they returned to Parthia? What all was returned? I Know I've seen maps where a large part of these conquests were still controlled by the Byzantine Empire up until the Muslim conquest. Were they simply reconquered or was only part of the area given up? Was Roman Mesopotamia relinquished simply because it was were too hard to administer, because from what I've read Trajan whooped Parthia pretty good and the Parthians don't seem as though they would have been able to recover enough to be a threat for at least a while.
Traianus, first of all, didn't really kill the Parthian state. He basically took advantage of a civil war and marched with scanty opposition all the way to the Persian Gulf. There was very little weight behind his conquest of Parthian Mesopotamia, and as soon as the Parthians finished their civil war they would be able to fully oppose the Romans militarily. In addition, he had left several sizable and defensible cities unconquered in his rear, threatening his lines of supplies. It was in besieging one of these, Hatra, that he ended up dying. Hadrianus correctly realized that the conquests were unsustainable and prudently withdrew from virtually all of them, returning the frontier to more or less the line of the Euphrates. Armenia was returned to semi-independence as well, with a Roman-installed puppet ruler in charge.
Several wars were fought between the Parthian state and the Roman Empire after the reign of Hadrianus. During the reign of Marcus Aurelius, for instance, his co-emperor, Lucius Verus, fought against the Parthians, a war which saw some minor successes for the Romans but which was interrupted by the famous Aurelian Plague. Septimius Severus, in the 190s and 200s, landed the most decisive defeat in Roman history on the Parthians in a more sustainable way, conquering essentially all of Assyria (province of Roman Mesopotamia) up to the great fortress of Nisibis. He sacked Ctesiphon as well. His son Caracalla fought another war in the 210s and had some military successes, but was assassinated. Caracalla's successor, Macrinus, was defeated by the Parthians and forced to come to terms.
After the Sasanians basically usurped the Parthian confederal empire, they ended up being more militarily protagonistic. Roman Mesopotamia was pared down somewhat in a series of repeated wars that saw Sasanian armies raid heavily in Syria and even into Anatolia at some junctures. Just as the Romans had taken advantage of Parthian civil wars to temporarily occupy Ctesiphon under Traianus, ābuhr I took advantage of the civil wars associated with the "Crisis of the Third Century" to sack Antiocheia. The Sasanians temporarily evicted the Romans from the eastern part of Roman Mesopotamia - though the Romans still maintained control of some territory on the eastern banks of the Euphrates. But the pendulum shifted again and the Romans smashed the Sasanians under Galerius and Diocletianus at the Battle of Satala in 298. In the ensuing peace the Sasanians gave up all the territory they had lost to Septimius a century earlier, plus were forced to devolve control of much of the actual frontier to local authorities, as a way of preventing the Sasanians from fortifying the area heavily.
Pendulum swung again after Constantinus I died in 337. His successor Constantius II spent basically his whole reign fighting against ābuhr II with mixed success. When Iulianus launched his little usurpation in 361, he inherited the eastern war. He promptly amassed a colossal army that marched to Ctesiphon, roundly beat up the Sasanian army in a field battle, and accomplished precisely nothing outside of that. We're not entirely clear as to why, though inadequate staff and supply arrangements on the part of the Romans, and perhaps
too large an army. Anyway, Iulianus was forced to turn around, got his dumb ass killed, and his successor Iovianus was saddled with the unhappy responsibility of surrendering Nisibis and Singara (two major fortress-cities in eastern Roman Mesopotamia) and withdrawing from Armenia. ābuhr II then invaded Armenia and divided it up with the Eastern Emperor Valens in the 370s.
Fifth century was mostly peaceful, apart from two crises that amounted to little more than border skirmishes. The Romans were busy with the Balkan frontier, and the Sasanians had to deal with the Hayāṭila of Central Asia. That all ended with the Anastasian War of the 500s, which inaugurated another century of Roman-Sasanian warfare.
This handy map shows the situation at the end of that century, in 600. It's a little skewed towards the Roman side, since the Romans had just scored one of their largest victories ever. Emperor Mauricius had employed a civil war in the Sasanian Empire (funny how that works), invaded to restore young Xusrō II To His Rightful Throne, and took a sizable slice of Armenia into the bargain. Mauricius got knocked off his throne by a revolt and civil war, which gave Xusrō an opportunity to invade and snag some loot, which kicked off the Last Roman-Persian War (602-628), about which I will write a history article eventually, which saw the Romans get the living crap beat out of them for twenty years, then an astonishing comeback (utilizing - what else? - a Sasanian civil war in part) that saw them regain all of the territory they had had under Mauricius.
In sum. The Romans didn't
really have the resources to kill the Parthians, and couldn't even score significant successes against them without employing Parthian civil war. Holding all of Iraq, like Traianus, tried to, was improbable and untenable. Western Kurdistan and Syria were reasonable goals, but more was unlikely. By the same token, though, the Sasanians were even less likely to score major military successes against the Romans unless civil strife or exogenous shock (like plague, or an invasion on another Roman frontier) intervened in their favor. Rome was clearly more powerful than the Parthians and Sasanians throughout their entire relationship, but not
that much stronger.
That answer your question?