There's a difference between parental responsibility and the people who gave birth to to a child. I think you are getting the two mixed up.
Hypothetically lets take two parents, they can be any sex, gender, sexuality you want. They have a child together. A while later they get divorced/separate whatever you want to call it, and one gets custody. The one with custody get a new partner. Does the child's birth certificate get changed to reflect the new partner's parental responsibility? If you are treating it as a parental responsibility certificate then surely that child's birth cetificate should be scrubbed of the old partner and a new partner added every time this happens?
Should a hetro married man with kids be able to go back and change the children's birth certificates if he decides he wants to pretend to be a woman later in life? "I felt like a woman at the time. I want my kids birth certicates changed."
In all these examples and what you said above the wants of the adults are assumed to be a priority over the rights and needs of the children. Surely this is wrong. The birth certificate is for recording the birth of the child, not reflecting the whims of selfish parents. The parents have their own birth certificates, let them fuck around with those if they want to play ego games.
Should an adopted child have their birth certificate wiped clean and the adopting parents put in just to keep the adults happy? Is that a good sign that the child will be loved and cared for?
Stop putting the petty selfish need of adults who are meant to be the mature ones in charge making the sensible decisions before the needs of children who are meant to be the ones protected by the adults.
If two lesbians love and care for a child why do they "need" superifical lies on official paperwork to be happy? If the truth on a birth certificate is an issue for a parent that parent is probably going to be a really shitty parent to the child.