Paints don't have to be half Quarter Horse. Paints are desended from quarter horses, they were the colored horses that couldnt be registered with the AQHA. Now adays, if a horse is to be registered paint, at least one parent must be a paint, the other can be a thoroughbred or quarter, etc.
Pinto's can be registered in the PTHA as long as they meet the white minimum and aren't a draft or a mule. Studs can only be registered if they are registered in a certian breed registry. Some horses that have paint parents can be registered even if they don't show paint markings.
Paint: APHA.Com - The Breed
Here is my paint:) He is registered APHA, and is a paint x quarter, but looks more thoroughbred.