I've never posted a response to a question before, but I see no one's touched this and I think it isn't a very difficult question to answer.

The Roman Empire was bordered in the East by the Parthian Empire, along the Mesopotamian frontier. Indeed, the Romans would be opposed in the East by a major power throughout most of their history.

As for what the Roman and Parthian fights look like, we know that in the spring of 53 B.C., Rome's richest man, Marcus Licinius Crassus, decided to use his command to fight a war against the Parthians. Goldsworthy, in Caesar: Life of a Colossus notes that Crassus was confident in his success, with a force of 7 Legions at his command. Ideally, these legions had around 5,100 soldiers. This varied, of course.

The armies were very different from one another. The Roman army was centered around their tough legionaries. Heavily armored and equipped with a shield and a couple of pila, they had generally done pretty well against every foe they had come up against. The Parthians experienced similar success against their foes, but their army was based around their two types of cavalry. First, the lance wielding cataphracts, with both the horse and the rider protected by heavy armor. Then, the composite bow armed horse archers.

They fought an engagement at Carrhae, which Goldsworthy argues wasn't as bad of a defeat for the Romans as is often claimed. Crassus' son, Publius, was drawn away from the main force and killed - but the battle was a tactical stalemate. The Romans had gotten the worse of the fighting, though, and were a long way from home. In any event, the night after the battle Crassus lost his nerve and ordered a full retreat.

It was in the retreat the Roman army was utterly destroyed by the pursuing Parthians. Crassus attempted to negotiate with the enemy, but he was killed and his head sent to the Parthian king.

Further, we know that Caesar was planning on departing for a military campaign against the Dacians and Parthians before his death.

Hope that answers your question. I could go into Augustus, but the general pattern for the future was on/off war between the two powers.

I used Adrian Goldsworthy's Caesar as my source.