Legal Question in Family Law in Texas

Before the father of the child has any legal rights, he has to be legally recognized as the father. There are several avenues to establish paternity. He also needs to petition the court in a suit affecting parent-child relationship to establish issues of coservatorship, support, possession and access.

I agree with the previous answer, but to answer your question, a father has exactly the same legal rights as a mother. If the parents can't agree, then it is up to the court to divide those rights, including the right of possession and the right and duty of child support. The particular facts of the case will guide the judge.