I was wondering is it permissible for an unmarried woman to move out of her parent's home and live by herself if her parents are still alive?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.

Imaam Ahmad reported that the Prophet prohibited staying alone; he prohibited for a man to spend the night alone or to travel alone. Since this is forbidden for a man, then it is more forbidden in respect of women especially if there is no reason for her to stay alone or to be in an insecure environment. Therefore, it is not permissible for a woman to move from her parent's house and live alone. If her parents are in need of her, then the prohibition is more confirmed regarding her moving away.

Allaah knows best.