The Romans used archers (and slingers), especially in the later stages of the empire. These were auxiliary troops at first.

Archery can be deadly but a Roman infantryman is well protected, and they also evolved the testudo formation for even greater protection against missiles. An infantry archer is effective (as long as he has ammunition) untill the Romans have closed and then they die. Another factor is the weather. A wet bow is useless so the dry climate of Mesopotamia and beyond is very friendly for archery warfare.

A cavalry archer is another matter. The Parthians, and later the Sassanids, perfected the light (and heavy) cavalry concept. This was a sophisticated operation. Mounted archer troops would move in, barrage the Romans with arrows until they ran out, then retreat for a fresh supply of arrows from a camel supply train. This kept the Romans effectively pinned down and slowly withering away.

Roman cavalry, never their strong point untill the Byzantine age when they became the equestrian equals of the Sassanids, was outnumbered and generally lured away from the main army and destroyed. The Parthians also evolved the deadly 'Parthian Shot' which allowed them to turn in their saddle and rain arrows on the pursuing Romans.

To make matters worse bow- and horse equipment technology evolved in time, giving rise to much more deadly archers, then armored heavy cavalry, and finally the classical 'tank', superheavy cavalry called cataphracts, the Sassanid Asvārān and their Byzantine variant, the Clibanarii. These would later evolve into the European knight.

After the endless rains of arrows (some calculations are talking about more than a million arrows fired in a single engagement) the Romans would be battered and their morale would be crushed, after which the Persian heavy cavalry would perform lance charges at weak points with devastating effect.

In the many wars fought between the Romans against the Parthians and Sassanids, battles on open terrain fit for cavalry were usually completely disastrous for the Romans, but in more rugged terrain like Armenia the Roman infantry could stand their ground. The Romans got their first serious taste of this new kind of warfare at the Battle of Carrhae in 53 BC, where 7 Roman legions (ca. 40-50.000 men) were completely destroyed at virtually no loss to the small Parthian army. Fortunes of war (but also cooperation) came and went, but the Sassanids came very close to destroying the Byzantine Empire completely in the Byzantine–Sassanid War of 602–628 AD

My primary reference for this is Shadows in the Desert: Ancient Persia at War by Kaveh Farrokh