In just the Republic years it had dealings with the Seleucids & Parthians. Were there any Roman mercenaries fighting with Xenophon? How many Persian Governments fell while Constantinople watched the bodies of its enemies float downstream?
As you mentioned, the Romans did fight against the Parthians pretty frequently (the Seleucids were Macedonian, though they ruled over much of the former Achaemenid Empire). The Sassanian Empire emerged in the early 3rd Century and was the last empire that could be considered truly Persian, they fought with the Romans pretty frequently as well. There were other states that fought with the Romans and held most of Persia such as the Rashidun Caliphate(Islamic Caliphate founded after Muhammad's death), Umayyad Caliphate(essentially the same entity with a different dynasty claiming to be distant relatives to Muhammad) and the Abbasid Caliphate (also claimed to be the relatives of Muhammad, overthrew Umayyad administration), but the three Caliphates were not Persian in the sense that they were ruled by Persians. You could maybe argue that the Seljuks and the Anatolian successor, the Sultanate of Rum/Konya were Persians, they were Turkish people but they heavily embraced Persian culture, particularly in the Sultanate of Rum where it was the administrative language. I have no idea about any direct Roman interactions with Persian states prior to the Parthians.
The Romans fought with two major Iranian dynasties: the Parthians, and their successor dynasty, the Sassanids. It's important to note that Persians are just one of many different Iranian peoples so to refer to the Parthians as "Persian" would technically be incorrect, as they came from Parthia in the northeast of Iran (modern day Khorasan) rather than from Pars, where the Persians came from (south central Iran). To be fair, the Parthians kept intact almost every aspect of Persian rule so its understandable why they might be considered "Persian".
The Parthians began hostilities with the Roman Republic in 92 BC. This continued almost uninterrupted through the Empire days. The Parthians were overthrown by the Sassanids, and hostilities resumed between the Romans and the Sassanids. When the Roman Empire split, hostilities still continued between the Byzantines and the Sassanids. Finally, after nearly 800s years of almost continuous war, both sides' resources were exhausted, paving the way for the Muslims armies to come through and take vast quantities of land from both the Byzantines and Sassanids--outright ending the reign of the latter for good.
Read more here: http://en.wikipedia.org/wiki/Roman%E2%80%93Persian_Wars