In AMHA, mares that are used for breeding have to be DNA tested as do the stallions so parent qualifications can be done if requested. Basically, they are proving a foal is really from the parents listed. Then, if a horse isn't registered within a year of birth (I think it's a year), they have to be DNA tested to be sure they are truly the offspring of the parents listed on the application.
Remember the craziness with Sweetwater who just ran stallions with mares all in one pasture, and then just "picked" the parents for paperwork. Then later in life if a horse was bred, it would be discovered that the parents on the paperwork WERE NOT the horses parents. It was a nightmare, and from what I've heard, the owner went to jail over registration problems and selling horses without knowing who the parents really were.
The AMHR does NOT require DNA testing.