Upvote:2
Wool breathes well. Central Asia can also be very cold. They didn't live in their armour. Consider how hot soldiers in tanks must have been in WWII in places like North Africa. People got used to it. The crusading armies were wearing gambesons and the like in their wars.
Armour needs to balance between protection and wearability. The heavy bronze age armours from Mycenean Greece for instance are thought to have been worn by elite warriors fighting from chariots so bulk wasn't an issue. A simple woolen tunic is enough to make armour comfortable and they could have used layers of linen too. The cataphracts used lances like medieval knights so I'd image they would want to have some decent padding under their armour. Being on horseback means that they're not using anything like the energy they'd be using if on foot so heavier clothing would be acceptable.
Consider too the Mongols who used a lot of wool and they wouldn't have given up their armour as they spread west through those hot arid parts of Asia. It took the heat and humidity of India to stop them spreading south.
Wool on a hot day is surprisingly bearable.