I'm not sure if APHA accepts Thoroughbred dams or sires, but they used to. My paint had all kinds of Thoroughbred bloodlines, but they were from generations past. In the case that they DO accept them, your horse could be a registered paint. Otherwise, no dice.
If he/she is colored like a paint, you can register him/her as a Pinto. They'll take anything with color.
But if you're looking for something to write down on paper, I'd go with "Thoroughbred/Paint cross," honestly =)
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