The Parthians had two major advantages the Romans initially struggled to counter:

1. Highly mobile horse archers who could stay out of range of the Roman legions while raining arrows on them. If infantry or supporting cavalry charged these horse archers, they could easily flee while firing over their shoulder, the so-called, "Parthian shot." This retreat often separated pursuers from their support, allowing the Parthians to sweep in and surround their exhausted attackers, often with...
2. Cataphracts: A cavalry force in which the horse and the rider were armored head to toe with scale armor or mail. They generally used a lance as their primary weapon, but also had maces or swords as sidearms. They were probably the heaviest cavalry the Romans faced up till this point, and the light and medium cavalry Romans fielded or hired to support their legions had trouble countering them.

The fact that the Romans went from being soundly beaten several times during their earlier conflicts with the Parthians to regularly marching to and sacking the Parthian capital, Ctesiphon, can be put down to three countermeasures which, when deployed together and used by disciplined troops, negate the Parthian's advantages. These tactics first became popular under Trajan, and were used for several centuries.

First, the Romans began using masses of infantry slingers and archers to outrange the Parthian horse archers and keep them at bay, largely neutering them. However, these ranged troops didn't have much impact on the heavily armored cataphracts, and would have been highly vulnerable if not used in conjunction with the second countermeasure.

This was the hollow fighting square. The idea was the create a hedge of multiple ranks of pikes facing outward in all directions so a force could not be flanked. Inside this hedge, Roman missile troops could shelter from attack. To pierce the square, cataphracts had to charge through multiple ranks of pikes and then face a withering barrage of heavy throwing spears and lighter javelins, as well as missiles from the archers and slingers.

The third countermeasure was increasing the number of medium and light cavalry the Romans brought with them on campaign. These units would shelter in the fighting squares, and once a Parthian force had been mauled trying to assault one, the fresh cavalry units would be unleashed.

When competently trained and lead, the Romans became very hard to beat with traditional Parthians tactics. The Parthians figured this out, and became reluctant to throw themselves into pitched battles; their efforts switched to attrition.

The Parthains would try to wait until terrain made it hard to form a square, or the Romans became strung out during a march. Or, as sometimes happened, some combinations of supply and communication issues drove the Romans back.

But by and large, the Romans were able to figure out how to counter the Parthians, and regularly beat them in the field and captured their cities.

The bigger issue was holding any significant portion of the Parthain empire, but that's another story.

Sources:

• Dio 40.1.5; 49.20.1–4.
• Goldsworthy, Roman Army op. cit., p.229.
• Plutarch, Antony 38, 42–3;
• Tacitus, Annals 13.36–40; 15.9.
• Sherwin-White, Roman Foreign Policy, p.315