Horse archers may be awesome, but they are not the end-all or a game breaker - at least against enemies who know 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 harassment was useful for disordering enemy formations, but the actual killing happened during hand-to-hand combat (where a disordered formation was more vulnerable). This is due to one simple fact: armour is not useless. In fact, against the relatively weak bows of horse archers, even cloth armour (gambeson) can be counted on to reliably stop arrows. At the same time, horse archers themselves have to use short bows which are outranged by the 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 a vacuum. They were used for harassment and to disrupt enemy formations before a heavy cavalry charge, but the decisive blow was always given by the heavy cavalry. At Carrhae, horse archers forced Romans to form testudo and thus make themselves a perfect target for a cataphract charge, but it was successive cataphract charges which broke Romans. Byzantines likewise used horse archers to "prepare" enemy infantry for a cataphract charge, attempting to disorder enemy spearmen before heavy cavalry crashed into them. And in the 15th century, heavy cavalry fought in speacialized wedge columns similar to the Byzantine model, which had heavy cavalry at the front and outside, with the inner ranks formed from mounted crossbowmen. The idea was same as that of Byzantine cavalry wedge: softening enemy resistance before a heavy cavalry charge hit, but again, missile fire was not supposed (or expected) to defeat enemy resistance on its own.
