A Father by any other name?

At common law, a child is legitimate when born during the lawful marriage of his parents. A child that is born to a married woman is presumed to be the child of her husband. Both parents have parental responsibility for the child. There is a presumption that the husband (“H”) is also the father of the child. What if, however, the child is fathered by (“F”) a lover or boyfriend? Who has parental responsibility then?