He dosen't run....he gallops!
The idea was to try and give the effect of horse archers shooting while on the move. In reality the Parthians were well known for shooting backwards while evading an enemy charge (this is where we today get the phrase "Parthian or Parting Shot"). After all, who would sit stationary on a horse while any enemy runs hundred's of yards/meters towards them? With a bow and a horse you can keep the enemy at a distance while still inflicting casulties; that's what made horse archers so good for so long; they can kill you but you can't even reach them.
However, I do admit it takes a little getting used to (have you tried two horse archer units fighting each other yet?) If you like, I can easily make a new less dynamic conventional civ3 "standing-still-shooting-forwards" FLIC instead of the "galloping-on-the-spot-shooting-sideways" one. It's no problem, and should only take a week. Let me know and I'll get started.