Personally I have distant African ancestry and I grew up with an African American uncle and thus spent time in the community. However, I'm quite aware that no matter how much I respect and appreciate black culture, I grew up with white privilege and I have not experienced their struggle. I do not have the right to claim to be black and I know that a vast majority of black people would be deeply offended if I tried to. If I really do feel a sense of solidarity, which I do, then I should place their wishes and comfort above my own (if I were tempted to pass as black somehow). That's what being an ally means. Yes race has been culturally constructed--by racist white people--which is all the more reason for me to respect black people's wishes in this regard. It's their culture, it's their struggle. I'm just a witness to it.
That said, her parents are absolutely creepy. If they ever had a desire to reconcile with her, publicly speaking against her is not the way to go.