Ok yes that clause does appear to cut Asterion off from holding the deed. Though the constitution itself lists his parents as Pasiphae and King Minos. As an adopted person myself I would count Minos as Asterion's human parent. Minos may not be his sire, but he is listed as his parent, and it is Minos's house that Asterion was born into.
Its a technicality that may not win out, but sometimes stories turn on technicalities depending on who is adjudicating.
even then, i think i recall the hades judges claiming that king minos lineage has always been part bull. and robert's ability to see souls tells us that Asterion has a hybrid soul. Asterion's soul is half chaos, though, according to robert, it's like that half was stripped away from him. When we see his forehead birthmark appear, i believe it to be a sign that he is recovering that half of his soul, the direction of the story seems to be towards him embracing poseidon's heritage.
the question to consider is what being born of human parents mean. does it implies no mythical could ever hold the deed? or would someone from like the panoptes lineage which is said to be human in origin be eligible for it? would it be a more literal interpretation and go that parents must be humans, and, as you posed, do these parents have to be the ones who conceived them or will the ones that raised overrule?