Your mare is 75% QH. I looked her up on allbreedpedigree. Her sire can't be double registered as AQHA because he's a tobiano. Tobiano doesn't exist in Quarter Horses; however, all of the "overo" genes do, hence double registration.
Double registration comes from when the AQHA had their white rule that banished all horses with what they considered "excessive white" to the APHA. However, the APHA changed their mind after a while and said that at least one parent had to be APHA, but still allows outcrossing to TB and QH. At that point, the AQHA allowed the horses that are actually AQHA horses registered as APHA to go back and be registered as QHs.
However, some Paints either don't inherit a white gene from their parents or just don't express it enough to receive full papers and are registered as Solid Paint-Breds.