Basically what Ink said. And yes, the APHA now requires at least one parent be APHA registered. However, I do believe that if that parent is AQHA/APHA, the foal can also be double registered if the other parent is AQHA.
Paints are their own breed because tobiano doesn't exist in Quarter Horses, but the two are ridiculously intertwined. Abby is APHA because her dam was APHA, but her sire is AQHA. She's also solid coloured, so she has Breeding Stock papers instead of legitimate APHA. If I were to show, we would be in a separate class just because she doesn't have any big enough visible spots. (Though I'm pretty sure if I found $25 to send off for a LWO test, it would come back positive.)
I'm not 100%, but while the Excessive White rule was in place, the AQHA sent its "spotted" Quarter Horses over to the APHA, but I don't think the APHA ever sent solid Paints to the AQHA. Don't quote me on that.
The Palomino registry and similar ones are just colour registries. They don't really mean much, IMO. Just a piece of paper stating that your horse is a Palomino and you could show those particular shows.