I read from wikipedia that Rome never managed to beat Parthia after the mind blowing defeat the Parthian General Surena inflicted on Crassus during the Battle of Carrhae in 53 BC, but then after being convinced that Rome backed down and never tried to venture east, I read that there is another Roman Emperor called Trajan who waged war on Parthia and even managed to overrun its capital Ctesiphon, and the aftermath of this war Mesopotamia, Armenia and Assyria were added to the empire as provinces of Rome. Then if Trajan did conquer Parthia and had the Roman troops in Babylon then why did Septimus need to lead another campaign against Parthia in the future, unless Parthia was never really conquered?
This is more or less entirely sourced from Wikipedia and from previous comments. I am not an expert and I encourage others to provide richer answers with better sources.
How much of the Parthian (Arsacid) Empire did Trajan conquer? Your question referred to the Roman Empire conquering "Parthia;" but this is not right. To be clear, Parthia is a region in Central Asia, quite remote from the frontier between the empires or the territory that Trajan invaded and occupied during his Parthian campaign. Trajan's armies invaded and seized the western territories of what's called the Parthian Empire, a much larger entity, which was ruled by a dynasty (the Arsacids) whose origins were associated with the region of Parthia. Parthia (the region) is located in the northeast of modern-day Iran and the south of modern-day Turkmenistan. Throughout the classical antiquity it was usually ruled by imperial dynasties based out of Persia and Mesopotamia, such as the Achaemenid Persian Empire (ca. 540 BCE-330 BCE) and the Sassanian Persian Empire (224 CE-651 CE). During the period that you're asking about, Parthia was known as the traditional homeland of the royal dynasty (the Arsacids) that had come to rule over a much larger empire in Western Asia, Mesopotamia, Iran and Central Asia. The dynasty was associated with Parthia because they began their rise to imperial power around the 240s BCE by first taking control of Parthia and then expanding outward from there.
The territories you mention, which were conquered during Trajan's conquests, Armenia, Mesopotamia, and Assyria, were very rich and important territories to the Parthian Empire. The fact that the Romans captured their southern capital, Ctesiphon in 116 CE (located on the eastern bank of the Tigris, near modern Baghdad) was an especially severe blow. But the Romans never conquered, nor attempted to conquer, the entire territory of the Parthian Empire, which extended far into the Iranian plateau and Central Asia. @PieterGeerkens notes in comments that the territory of the empire was perhaps three times as large as the territory conquered during the invasion.
How much did it matter that the Romans overran the capital at Ctesiphon? It's important to remember that "capital cities" in ancient empires the did not work the same way that they work in modern nation-states. And especially not in empires like the Parthian Empire, in which the ruling dynasty explicitly held on to many traditions and practices of rule connected with their ancestors' roots as nomadic warriors. The Arsacids (like the Achaemenids before them and the Sassanids after them, etc.) had multiple "capitals" simultaneously. "Capital" here means not a special seat of government with a unique status over the provinces (like the city of Rome or, later, Constantinople), but simply a big city with a royal residence, and usually also some other institutions of government, e.g. a royal mint. These included not only Ctesiphon and Seleucia on the Tigris but also Ecbatana, Rhages (Ray), etc. Ancient Iranian Shahs like the Arsacids typically moved around between their capitals, and the ceremonial seat of government traveled wherever the Shah himself happened to be. Conquering one of these cities in 116 did not, indeed could not, have the same effect that, say, capturing Paris would have in 1871 or 1940. There was no single point on the map that Roman armies could capture which would turn over control of all the territories, or all the vassal states that maintained Arsacid rule, to the Romans.
Capturing the person of the Shah himself might have had an effect like this, if control over the Empire were undisputed; but Romans never did this, and in any case the Parthian Empire was already in a state of dynastic conflict at the time that Trajan invaded. The western territories that he conquered had been mostly controlled by Osroes / Khosrow I, who was temporarily deposed after the loss of Ctesiphon, but the territory controlled by his rival Vologases / Walagash III was never seriously touched by the invaders. Some Parthian forces had to retreat from their western territories and their westernmost capital, but they still held a huge territory when the war ended.
How long did the Romans hold onto their conquests in the Parthian Empire after Trajan? Trajan may have contemplated conquering further -- Roman armies continued eastward from the Tigris and reached as far as Susa -- but he died (in 117 CE) and his successor, the Emperor Hadrian, was not interested in continuing the war. In fact Hadrian immediately returned control of Ctesiphon to the Parthian Empire in 117-118 as part of a peace settlement with Osroes I. He quickly abandoned the remaining conquests east of the Euphrates in a peace treaty. Hadrian's reasons for doing this may have been driven by any number of considerations; during his first few years as Emperor he seems to have both been very concerned about the stability of his own rule (Hadrian's Praetorian Prefect carried out a purge of a number of military commanders who were accused of conspiring against him), and also about the military situation in both the new conquests and back in the rest of the Empire. Major revolts broke out in Judaea and Babylonia, and there were threats along the Roman Empire's other frontiers, all of which demanded military attention, and which made ongoing control over the new conquests hard to maintain. By the 120s the Parthian Empire was in control of almost all the territory that they held at the outbreak of the war.
To conquer a place, you to have to annex all of it to your realm. The Roman Empire never annexed all of the Parthian Empire, so no emperor, not even Trajan, conquered the Parthian Empire.
Here is a similar situation from history.
In 1805, Napoleon invaded Austria, leaving detachments behind at various points to guard his supply route. Napoleon occupied Vienna, capital of the Austrian Empire. The Austrian army and government retreated, and Napoleon pursued them, leaving various detachments behind to guard his supply route, into the Kingdom of Bohemia which was part of the Austrian Empire. A Russian army, allies of Austria, arrived in Bohemia.
The Russian General Kuzutov argued that the allies should retreat farther east into the Kingom of Galicia, also part of the Austrian empire, to link up with Russian reinforcements on the way and to stretch out Napoleon's long supply lines even farther. But Tsar Alexander I wanted to fight, so the allies fought Napoleon at Austerlitz and were defeated.
After Austerlitz, the Austrians made peace with Napoleon and had to give up several provinces to Napoleon and his allies. But Napoleon didn't annex all of the Austrian Empire to the Franch Empire. Thus Napoleon didn't satsfy the condiitons for saying that he conquered Austria in 1805.
Similarly, in the war in 1809, Napoleon invaded Austria, leaving detachments behind at various points to guard his supply route, and occupied the Austrian capital Vienna. The Austrian army and government retreated, and Napoleon pursued them.
The Austrians under Archduke Charles defeated Napoleon at the Battle of Aspen-Essling on 21-22 May, 1809, but were defeated at the Battle of Wagram on 5-6 July, 1809. After Wagram the Austrians made peace with Napoleon and had to give up a few provinces to Napoleon and his allies. But Napoleon didn't annex all of the Austrian empire and so didn't conquer The Austrian Empire.
In 1814, The Austrian armies and their allies invaded France. As earlier, Napoleon kept defeating the allies. But unlike before, each time he defeated the allies they were closer and closer to his capital Paris. When the allies reached the neighborhood of Paris, the French politicians turned against Napoleon and forced him to abdicate, because Napoleon's regime was much less secure than the Austrian Emperor's was.
So a politically insecure regime will often fall if its capital is captured or even threatened, but a politically secure regime can stand and continue to fight even after its capital and heartland is captured by the enemy.
And the French-Austrian situation during the Napoleonic Wars is similar to the Roman Empire invading Parthia and capturing the winter capital Ctesiphon, and in later centuries invading the Sassanian Empire and capturing the capital Ctesiphon. This happened several times over the centuries, but the Romans never conquered all of the Parthian or Sassanian Empires, which included most of modern Iran and usualy some lands beyond Iran to the east.
So the Romans never conquered the entire Iranian Empire. Sometimes the Romans conquered forts, cities, or provinces from the Iranians, but they never conquered the whole Iranian empire. And sometimes the Iranians conquered forts, cities, or provinces from the Romans, but they never conquered the Roman Empire. Khosrow II (r. 590-628) came close to conquering the Roman Empire but the war turned out disastrously for him.
Trajan did better than any other Roman invader of Persia, and better than Napoleon did in his Austrian Wars. Trajan not only occupied the Partian capital Ctesiphon in Mesopotamia (modern Iraq), but he declared that Mesopotamia, included Ctesiphon, and Armenia were annexed to the Roman Empire.
To equal that, Napoleon would have had to have annexed the lands of the modern Republic of Austria, including Vienna, to the French Empire. But just as Trajan didn't annex the Iranian lands of the Parthian Empire, if Napoleon annexed Austria but only Austria, the Austrian Empire would have been left with Bohemia, Galicia, Hungary, etc. and would still have been a major power.
I remind you that the founding city and original capital of the Roman Empire, Rome, was captured and lost to the Roman Empire in the 5th century AD, in 410, 455, and finally in 476. But the Roman Empire still controlled its eastern section - that is misnamed by historians the "Byzantine" Empire - from the new capital of Constantinople for a thousand years of expanding and shrinking in size, until 1453. In one of the phases of expansion, the eastern section of the Roman Empire even recaptured Rome in the 6th century and held it for about two more centuries.
So even though Trajan conquered two provinces and the capital city from the Parthian Empire, the Parthians still ruled a vast region in modern Iran, just as the Roman Empire would later rule a vast area after it lost its original capital Rome. The Parthian Empire still had a lot of military power.
Trajan died a year after his victory over Parthia, and the new Emperor Hadrian abandoned Mesopotamia and Armenia, which allowed the Parthaisn to reoccupy Mesopatamia and Ctesiphon, and restored Armenia to the status of a vassal kingdom of Rome and/or Parthia.
So the Romans never conquered the Parthian Empire.