A Paint horse, in order to be registered, can have both parents be registered APHA paints, or one parent registered APHA. If only one parent is APHA, the other parent has to be a "stock breed," AQHA, or Jockey Club registered TB.
A Pinto horse, in order to be registered as a Pinto, simply has to have the markings, Despite that, most people consider pintos to be anything that has the markings but is NOT eligible to be registered APHA as a paint.
Appies can not be registered AQHA, even if they are just breeding stock (not colored). Neither can paints. But, if a horse who has two parents registered AQHA were to somehow pop out with appy markings or some white, he could be registered as an appy or a paint as appropriate. It was only a year or two ago that a horse who did happen to be born with some white between his nose and his knees can be registered AQHA. Prior to the new rule, they were out of luck even if both parents were registered.