Frankly, I don't care how many birds you do or don't have, or what your plans are. I'm interested in addressing the matter of DNA testing. I know it is possible to DNA test birds(the only way that I am aware of is a blood test), but I have no idea whether anyone has come up with any kind of test that is available to the public.
DNA testing is commonly done in some animals, and is actually a requirement for registration for some breeds. In the case of horses, you send in a hair sample. It isn't the hair itself that is tested, it's the little tiny clump of tissue that sticks to the "root" of the hair when it is pulled out. When the test is done, it compares the sample to samples taken from the (supposed) parents, and says, "yes, there is a good probability that this animal came from this parent," or "it is highly unlikely that this animal came from this one!" You have to have the parents' DNA "fingerprints" on file, the person doing the test then compares the baby's DNA to that of the adults, and decides whether there is enough similarity to make it likely that they are related.
You can't just take a sample and expect it to tell you who the father is, you have to have a sample from the possible fathers to compare the sample to, and then pick the closest match as the likeliest one. If the possible fathers are closely related to each other, you may have more than one that the tests say are probably the father. The same is true for the mothers. And if you have a mother in one group that is related to a father in another group, and vice versa, the test probably wouldn't be able to tell you which pairing the youngster came from, just that it was closely related to those potential parents. Unless a large amount of control is exercised over just who has access to whom, the exact pedigree of any particular offspring can't be known for sure.