Horse archers may be awesome, but they are not the end-all or a game breaker - at least against enemies who knows how to counter them. Ranged units in general are historically often overrated: English longbowmen could not have won any battle of Hundred Years War on their own with just arrows. Missile harrassment was useful for disordering enemy formation, but actual killing happened during hand-to-hand combat phase (where disordered formation was more vulnerable). This is due to one simple fact: armour is not useless. In fact, against relatively weak bows of horse archers, even cloth armour (gambeson) can be relied on to reliably stop arrows. At the same time, horse archers themselves have to use short bows which are outranged by longer bows (and later also crossbows) used by their infantry counterparts. As a result, if infantry has sufficient missile troops, horse archers will lose the exchange - as Romans proved after adapting to Parthian cavalry tactics.

This is why horse archers were never used in vacuum. They were used for harrassment and to disorder enemy formation before heavy cavalry charge, but decisive blow was always given by the heavy cavalry. At Carrhae, horse archers forced Romans to form testudo and thus make themselves perfect target for cataphract charge, but it was successive cataphract charges which broke Romans. Byzantines likewise used horse archers to "prepare" enemy infantry for cataphract charge, attempting to disorder enemy spearmen before heavy cavalry crashed into them. And in 15th century, heavy cavalry fought in speacialized wedge columns similar to Byzantine model, which had heavy cavalry at front and outside, with inner ranks formed from mounted crossbowmen. Idea was same as that of Byzantine cavalry wedge: softening enemy resistance before heavy cavalry charge hit, but again, missile fire was not supposed (or expected) to defeat enemy resistance on its own.