In the state of Louisiana can a father give up his parental rights to a child without the mothers approval

Not even WITH the mother's approval. If you look through the hundreds of threads here that ask about relinquishing rights you will learn that a TPR is extremely hard to get. Generally a court will grant a TPR ONLY to clear the way for adoption or if the parent is a danger to the child.

The father can stop coming to see the child, never call and never send a card. But he can not stop sending that child support check.