The Romans sacked Ctesiphon, the Parthian capital, four times, three during the Parthian period. However, the Romans already had a massive imperial state and were not terribly interested in adding new territory. Trajan had one of the very few attempts to actually conquer Mesopotamia (not Persia, however), but after sacking Ctesiphon the administration was bungled and the attempted Mesopotamian broke off.

By and large, during the first century where we have the best information neither the Romans or the Parthians were terribly concerned with aggressive action against either other, and their conflicts were largely over Armenia, as imperial rhetoric from both meant neither were willing to give the region up. Nero's clever solution was to allow the Parthians to choose a successor (as the Armenians themselves leaned towards Persia), who would then travel to Rome for coronation.

My understanding from Plutarch is that Julius Caesar intended to campaign in Persia, but his assassination stopped that. Crassus (who was not a general of the same stature as Caesar or Alexander) did invade Persia, but that went terribly. Marc Antony also tried to invade, but also failed (although he survived). Augustus subsequently made peace with Persia in exchange for the return of Crassus's captured standards. The Empire was pretty much done with wholesale conquest at that point (except Britain, Dacia, and some parts of Germania). Much later, Trajan had success against Persia, but the gains were temporary. And late Rome was a basket case (crisis of the 3rd century, etc.).