Legal Question in Family Law in Texas

Generally, a person does not pay child support after his parental rights are terminated. Termination of parental rights (generally) extinguishes the ongoing child support obligation; it does not eliminate a child support arrearage.

It is legally possible for a terminated parent to have an ongoing child support obligation, but I've never seen such a case.

He can't "give up his parental rights" unless a Court approves it, that is, unless a Court decides that it's in the child's best interests for the father's rights to be terminated. I have seen two separate cases where a child's parents agreed to terminate the Dad's parental rights, but the Judge said "No."