Yes, Roman armies faced armies primarily composed of horse archers more than once. The fellow under me has referenced Attila the Hun. The Romans did technically defeat him at Chalons, but I'll admit to a small working knowledge of the late Roman army, so I'll focus on the earlier armies of the Republic and the Principate, and how they fared against horse archers.

In this time period, the two best examples I can come up with are both involving the Parthians. Crassus was famously defeated by the Parthians at Carrhae, and Antony led a failed invasion into Parthia while a triumvir.

The Parthian army was based on a feudal model, controlled by the king but in reality a large formation of many different strong noble families. Most of the army was composed of cavalry, mainly horse archers along with a smaller force of heavily-armed cataphracts. For example, the force of Parthian commander Surena that fought Crassus at Carrhae in 53 BC was made up of 10,000 horse-archers and 1,000 cataphracts. Parthian armies also included infantry made up from the poor, but these were never influential in any way at all and so have little tactical relevance. According to Lucian the basic unit of the Parthian army was a dragon of 1,000 horse-archers. The decimal model is quite prevalent in horse-archer armies, but we have no other evidence for this claim, and Lucian was no historian.

This cavalry force obviously operated in a very different way from Rome's infantry based army. We get a glimpse of Parthian tactics at Carrhae. In this battle, both sides expected an easy victory. The Romans had crushed every other eastern army they had faced with ease recently, and the Parthians were equally scornful of the Romans. Surena expected the Romans to be scared at the sight of his cataphracts, but was disappointed by the disciplined Roman infantry, who showed no signs of fear.

It seems that usually the Parthians simply expected the sight of their cataphracts to scare the enemy, and apparently they wore cloaks over their armour that they would discard right before battle, either as protection from the sun or to shock the enemy with a startling reveal of their glimmering scale armour.

Therefore, at Carrhae Surena had his horse archers bombard the Roman troops all day. Despite thousands of arrows, the Roman troops suffered relatively few casualties, mostly wounds, due to their armour and shields. Their morale stayed intact as well.

The Romans held up very well in spite of the horse-archer bombardment, and the troops were comparatively safe as long as they maintained discipline. However, Crassus made an ill-fated attempt to drive away the horse-archers, led by his son, who was cut off and killed. Dispirited, Crassus ordered a retreat, which soon degenerated into a routing mob as the horse archers surrounded the Romans, who began to panic. Carrhae was a decisive Roman defeat, and it made the Parthians very confident in themselves and scornful of the Romans. However, the Parthian army is often over-estimated in our sources, and they weren't truly as powerful as they would have liked to think.

Crassus's problem was that his army was unbalanced. Later Roman encounters with the horse-archers remedied this. For example, Antony brought many light missile troops with his Parthian expedition. A foot-archer will always out-range a horse-archer, and Antony's troops suffered little from horse archers. When the Romans were well prepared, usually the Parthians had to content themselves with shadowing the Roman marches, and depend on attacking their extended supply train. When Roman troops had enough supporting missile fire, they were very safe from horse archers, who became more of an irritant than a real threat.

Rome did sometimes have horse-archers in her armies, but it was never a major part of them in this period. I know that later Roman armies included more horse archers, but like I said I don't really know enough to be any authority on that. However in our period Rome tended to bring auxiliary troops into her army based on the native's methods of fighting, so many Roman armies in the east had horse-archer auxiliaries. For example, Arrian's army that fought the Alans, a steppe people, had horse archers in it. However they were only really a supporting wing and the Romans never adopted this tactic beyond in a minor supporting role.

I hope that this makes sense, I can elaborate more if need be. I'm not very good at organizing my answers on this subreddit yet, haha.

Sources: Plutarch's Life of Crassus & Antony, Adrian Goldworthy's The Roman Army at War 100 BC - AD 200