I don't know what all branches of the service demand with regard to your question, but I do know word for word what the ARMY requires.
When my 21 year son, married and the father of a 2 year old, decided to join the ARMY, he and his wife were told that HE had to give up 'parental rights'. They were upset by this b/c they are happily married. This was always something he wanted to do, she supports his choice, and they are happy!
So we MOTHERS of these two called the recruiter and found out that it is 'custody' that must be given over to someone, in the kids' case, his wife. We were also told that it didn't matter if he was married or not, this is how it was. It has nothing to do with the parent AS a parent personally, but more to do with the fact that the enlisted person now 'belongs' (as in ownership) to that branch of the service, and his/her first concern and obligation must be to that entity. It sounds frightening, but it is only (if so chosen) until the man/woman fulfills their term of enlistment.
Quote from page 2 of the Army enlistment contract:
"d. I am the parent of ****** and certify that the child(ren) has/have previously been placed and is/are in the custody of the other parent or another adult by court order. I further certify the custody agreement was and is intended to remain in full force and effect during the term for which I am now enlisting. I understand that if I regain custody of this/these child(ren), either by court decree, or in accordance with applicable state law, or if the child(ren) is/are residing with me in lieu of the legal custodian, I will be processed for involuntary separation for fraudulent entry unless I can show that the regaining custody is not contrary to the above stated intent; eg. death or incapacity of other parent or custodian)."