There is a logical reason why you are not legally entitled to receive an award of judgment against "dad" for the unpaid medical bills. You are not the owner of those accounts receivable, the care providers are the owners.
In any event it would not prevent the care providers from recovering a like money judgment against one or both of you. You may find some oddities in the law, but they are not that odd!
Your attempt to compare such a judgment with the family court awarding legal fees is not analogous. The difference is that in the latter the court is exercising a judicial discretion afforded by statute. And there are no statutes or case precedents authorizing a family court to award one spouse/parent a judgment against the other for an obligation owed to a third party.
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