The horse is trained to go from point A (beginning of parade route) to point B (end of route) WITHOUT the need for the rider to actually ride him. In fact, with the nature of the costume, the rider does very little in the way of influencing the horse in any way. So if the rider fell off, the horse would continue on to the end of the route as he had been trained to do.
As to the previous posters "glue factory" comment. First, gross. Second, not even remotely likely.
True! I did the KTTK tour several years ago and the subject of MNSSHP came up. The tour guide said that actually the horseman can see very little, if anything during that ride. The horse is trained very well to run from point A to point B. The horseman just has to hold on tight. He never actually guides the horse.