It is a complex question. The Romans used several methods to deal with Parthian horse armies.

First of all, they would build fortified camps, not just for their own troops, but also for the supply trains. A chain of forts, with stockpiled supplies for be erected in the wake of a Roman army in the east, to avoid being cut off from supplies or water.

On the immediate battlefield, archers and slingers were used. In ancient times, the sling often had a longer range than the bow, and cast led ellipse-shaped bullets could have a devastating effect on enemy troops, armoured or not.

The Romans attached vast amounts of Auxulia to their Legions for fighting in the east. Macedonian and Rhodian slingers and Cretan (often referring to a Greek way of skirmishing with bows that came from Crete rather than people actually from the island of Crete) and Syrian archers were renowned for their skill. It was easier to recruit, organise and supply archers and slingers on foot than on horse. Since the Romans often could bring foot archers and slingers in greater numbers than the Parthians could bring horse archers - and since the slingers often had a longer range, the Romans could defeat Parthian horse archers that way.

The Romans also recruited the service of the Saramatians, able horsemen from western Scythia (today roughly the Ukraine) and the Armenians, both whom fielded distinguished horse archers and light cavalry to bolster the cavalry element of their Legions.

To clarify, do you mean "how did they defeat Parthian forces in battle?", or "how did they defeat the Parthian Empire as a whole", because the Parthian empire remained unconquered by the Romans.

From my research, it looks like much of the fighting was over and in the buffer state of Armenia which is rougher terrain than the flat plains of Mesopotamia. How much did terrain factor into the Roman battlefield dominance?

[edit: clarity]