Even if the pups can't be shown, you can still register them. Even my pup who is the "correct colors" can't be shown because he's pet quality. (2 words: Afro hair
)
I find it odd that breeders claim lineage to AKC dogs, but then can't produce AKC papers for the dogs in their litter. I'm not saying this is your situation, rather something to keep an eye out for. The whole point is, even if they are parti yorkies, they should be able to be registered anyway. Otherwise, I'd be suspicious that someone somewhere bred dogs with AKC papers even though the breeder didn't have the permission to breed. Therefore, all offspring from that breeding cannot be registered.
Sorry if I sound jaded! So many people on YT, including myself, have been had when getting a dog. I personally live my dog, but we struggled so much at he beggining. Be informed, and research, research, research.