Just got the game last Thursday. While looking for a breeding pair that produced the least amount of red boxes I noticed that there is three different sized horses

Hiding Clever, Judge Concerto and Sticky Enjoy are all smaller horses, while Mosquito Forever, Turning Mariner and Dark Machine are of the larger variety.

When selecting a horse for breeding you can use the L/R buttons to scroll through horses. Mostly they are the same size, but you will notice that occasionally a horse will either shorter or taller.

Also observed that you are not obligated to accept the first foal you are shown. You can rebreed the parents until you find a suitable foal to race without having to continually reload.

8)