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Are you seriously asking what the justification is for an enforceable duty parents have to their young children?

Let's consider enforceable duty for fathers.

Say a woman has a man over to her house as a guest. The man asks if he may plant a few acorns in her garden. She says fine. He plants his pocketful of acorns in her garden and goes home.

Who is responsible for any resulting oak trees? Is she entitled to compel him to pay for any costs associated with the resulting trees? Not without a contract.

Does the man have an equal right to the disposition of her garden once he's planted acorns there? Of course not. Is he entitled to retrieve the acorns or the resulting trees after planting them? Not without a contract.

Consensual sex is not a contract to raise children, any more than the consensual planting of acorns in a private garden is a contract to raise trees. In the absence of a contract any responsibility for the result of consensual activity lies with the owner of the garden.

So what exactly is your moral justification for compelling fathers to pay for children for which they have not contracted?