According to Gov website :-
"Warning: Any person who deliberately gives false information for the registration of a birth may be prosecuted. Important Information :-
Parental Responsibility is acquired automatically by the mother and, where he is married to or in a civil partnership with the mother, the father. A father, who is neither married to or in a civil partnership with the mother, will acquire parental responsibility when the birth is registered or re-registered to include his details after he and the mother have acknowledged his paternity. Once parental responsibility is acquired, only a court can remove it. If you require further information, you may wish to contact Family Lives on 0808 800 2222 or seek legal advice."
Re-registering seems to be a bit of a minefield, so I suggest you follow the advice from ummmm and jno, or try the Family Lives number above.