I can see a brief tree with Karen’s birth and death in 1957.
Just to say, Michael, that the tree owner has named a father but that is not Ancestry which has listed him. It is the tree owner who has posted that information.
As has been said, the birth shows Cook as mother’s name. I haven’t seen the sister’s birth but if that shows the same name, then it is likely that there will be no father’s name on the certificates as the girls would be illegitimate.
You would then need to determine if the cousin is from the mother’s side (Cook) or the father’s side (perhaps unknown at the moment).
That is how I am reading your post anyway
