It would be perfectly feasible to devise an electronic form with various fields which would only be visible as required, and labelled as apt.
The simplest would have mother (father unknown). Birth parent, and one person with parental responsibility.
The commonest would have mother and father - birth parents autocopied to the fields for adults with parental responsibility.
Lesbian couple - mother. Father (if known ie the sperm donor). But the other woman fills the second parental responsibility slot.
And so on, to account for egg donor, gestational mother, heck even mitochondrial donor .... all the relevant information that the child might need.
The biological parentage should be unambiguous fact, as far as it's known, but the 'parental responsibility' part could allow a transman mother to be shown there just as one of the parents.
That sort of thing, but always with the need of the owner of that document ie the child first and foremost.