A parent... does NOT just abandon their child, because she runs away.
The Mom saying that the daughter "just needs to move out...." is in essence the Mom... abandoning her child. And the Mom not doing anything to HELP her child.... by means of whatever she can do or must do. ie: getting Therapy for her daughter AND the family, being a better parent and/or taking parenting classes, finding out WHY the daughter keeps behaving this way, talking to the school to see if the Counselors can help them and their daughter, etc.
Sure school is not in session now... but even during summer time, there is an office staff there. The parent can contact the school, speak to some Administrator, and maybe that Administrator can try and contact the schools' Counselor... even if just to find out RESOURCES of where the Mom can get help, or from who, or any shelters that are in that city etc.
Sending kids to a "boarding school" is also NOT the answer. That is just making someone else, the "parent" for THEIR child.
I went to a private boarding school when I was a Teen. A very exclusive private boarding school. And let me tell you... the school is not a "substitute" parent. Most of the parents that send their kids to boarding school... were not even involved with their kids' lives. They left it up to the school. And these kids... were NOT "angels."
This girl's parents... have to, do whatever they can... to HELP THEIR DAUGHTER, before something unthinkable happens to their daughter. If something happens to their daughter... they will never forgive themselves. Or.. they will perhaps, just see it as their daughter's fault. Not theirs. And that is PATHETIC.
The girl is a child... a parent has to intervene.
This is about their daughter's LIFE.
This girl's parents... do not even seem to "want" to do anything.
Because if they DID want to help their child... they would be scrambling like crazy to get their child help.
AND they would be getting Therapy for themselves, and/or attending parenting classes. At the, least.