Is it permissible for a woman to accept her parents’ spending on her without spending from her own small amount of money, and is her asking them for it considered asking for something forbidden, and is this spending considered a gift, with her parents’ consent to that?

If the parents are content to spend on their daughter, there is no impediment to that. However, it is not permissible for the daughter to ask for money without need, because that is from the blameworthy asking. The money they spend on her when she is not in need of it is considered a gift and a grant, and not from the obligatory expenses. It is forbidden for a father to show preference among his children without need.