Get this, they breed a homozygous stallion, (meaning he always throws his color, being overo or tobiano) to a SOLID Quarter Horse mare (or solid paint mare that they test for the overo gene to make sure she doesn't have it, if the stud is overo). A paint horse is just a colored Quarter Horse.
The breeder down the road from me, he has a couple hundred mares he breeds and they are nearly all solid. His babies are gorgeous colored paints. This man's got it going on.
I think the paint horse started out as a different registry and the AQHA appended it's rules to allow paints, but I'm not absolutely sure and I've forgot my PIN and can't access the AQHA web site.
My old stud is registered with Paint Horse Association. He's a red roan.