From what little I know, the horses were drilled and trained daily (especially in the ancient world), and were bred as war horses. The rider would have a connection with his own mount and I believe a Parthian commander was said the believe that "drills should be bloodless battles and battles should be bloody drills". The horses would be used to the sounds and smells of battle, and would have had years of training before being taken into battle. Cavalry was also quite often used on the flanks of an army, which meant there was more space for the horses, and there wasn't such a crush of men on each side.

Have a look at some of the Cavalry accounts from Parthia, Alexander the Great and the Dacians. The Parthians especially used cavalry as the main driving force in their army, and these were the people that delivered a crushing victory over the Romans at Carrahae.

Might be wrong on a few points, and my only knowledge is ancient history rather than more modern accounts .