During the Roman period, there are a number of different kingdoms that existed in the lands of the old Achaemenid Persian empire. As such, this will be in very very broad strokes.

Initially, there was the Diadochi, who were the successors of Alexander's empire. The largest of these was the Seleucid Empire, which stretched from central Anatolia to the border of India. The Seleucid Empire lasted until 63BC, when Syria was annexed by Rome, with the east of the country having been conquered by the Parthians prior to this.

The other major successor was Ptolemaic Egypt, which covered from Cyrenaica to the Levant, and down into Nubia. It often fought against the Seleucids to the north-east. It lasted until 30BC when it was annexed by Rome following the defeat of Antony and the death of Cleopatra.

The other really notable Hellenic state in the old Persian Empire was Pontus. This kingdom at its height covered all Anatolia and the ruler at this time was Mithridates VI (now often nicknamed "the Poison King" for his great fear of being poisoned), who was a major thorn in the side of Rome. The Romans fought three wars against Mithridates, which ended when he committed suicide in 63BC. Most of Pontus was annexed, although some remained as a nominally independent client kingdom, until it was annexed officially in AD62.

The real successor to the Achaemenids though were the Parthian and Sassanid Empires. They covered an area from the Euphrates and the Roman border to Afghanistan. These states, one after the other, provided a constant enemy to the Romans, which made the eastern border the most heavily defended along with the Rhine frontier. Their primary source of income was trade from along the Silk Road between Rome and China. The Parthian Empire originally came about when a people called the Parni invaded and seized the north-eastern portion of the Seleucid Empire in around 238BC. However, Parthian power didn't really kick off until the Romans broke Seleucid power at the Battle of Magnesia in 190BC. Then, in 148BC the Parthians began conquering the remnants of the Seleucid empire, which continued until the territory was under Parthian control in 129BC. Many wars were fought between the Romans and the Parthians, and indeed, sacking Ctesiphon (the Parthian capital) was nearly a right of passage for Roman emperors for a time. The main source of conflict between the two was Armenia, which was a buffer state between Rome and Parthia. At times both empires tried to annex Armenia, or to put their own ruler on the throne, with the other empire opposing them. In 224 there was a civil war in Parthia, and a new dynasty created a new empire, which is called the Sassanid Empire. It was territorially almost identical to Parthia for most of its existence. It was however, more hostile to Rome than Parthia had been. In a war against the Byzantine Empire in 602-628 the Sassanids conquered nearly the entire east, including Egypt, the Levant, and into Anatolia all the way to the coast of the Black Sea, before being repulsed by the Byzantines. At the end of this war, Sassanid power was broken forever, but the Byzantines were also exhausted, leading to the entire Sassanid Empire and the Byzantine Empire up to Anatolia being conquered by the Rashidun Caliphate in the following years.

As I said, extremely broad strokes, but it should give you an idea of who the major players were during the Roman period.