My ex-wife want to give up her paternal right to get out of paying child support. Do both parents need to sign something agreeing to this?

It doesn't work like that.
Relinquishing parental rights DOESN'T stop the maintenance payments being due.

The only things that do are re-marriage, adoption and death.

Have a read of this sticky: https://www.askmehelpdesk.com/family-law/signing-over-rights-read-first-116098.html

So you are saying you don't want the child support either?

There is nothing you can sign to get out of the court order or dead beat parents would not be dead beat they would be parents without parental rights.

If you do not wish to receive the child support money why don't you set up a trust account for the child(ren). You could even possibly help her out financially if she needs it. Once you have the money it is up to you what you do with it but you really need to apply it mostly toward the child(ren)

There are thousands of threads in this forum about this issue. A few minutes browsing through them would have gotten you your answer.

Only a court can terminate parental rights. Doesn't matter whether both parents agree or not. Courts are very reluctant to grant TPRs. Generally, TPRs are granted only to clear the way for adoption or if the parent is a danger to the child.

State laws vary. Some terminate rights AND responsibilities (like support) others only rights. But in states where responsibilities are also terminated, its even harder to get a TPR. Courts will not grant a TPR just to allow a parent out of paying support.

Ok, first unless there is another partner for the parent with custody, you can't just "give up your rights"
In some cases if both agree, it has been allowed but normally they still owe support,

If one parent could just give up rights and not pay support, there would not be 1000's of dead beat dads or moms in prison tonight for not paying.