In American civics, we have a clause int he Constitution that says that the States have to give “full faith & credit” to the laws of other States, and that only Congress has the power to make exceptions to this. That is why Congress passed DOMA under Bill Clinton, to prevent the gay marriage of one State to be recognized in another.
So the upshot is that the Courts will deny this suit because there has been no act of Congress yet granting the State of Louisiana the right to deny full faith & credit to the adoption laws of the State of New York.
In a sense DOMA is irrelevant, as this has nothing to do with marriage or civil partnerships. The issue is what a certificate of adoption should say, and clearly what it should name two (if there are two, and sometimes there won’t be) adopting parents e.g. John Doe and Jack Boe or Mary Doe and Martha Boe and should desist from using mother and father, although if there is an optional box, I see no reason why adopters should not tick whatever description they choose to tick.