Will parents insurance cover damage from an accident caused by daughter still living at home but not on their policy?

If she was driving their car, with their permission, they will probably pay the claim. But it is a certainty the rates will increase substantially. They may deny the claim. Many auto policies require that all licensed drivers in the home either be listed on the policy or be specifically excluded. Failure to list the daughter denied the insurance company their fair premium for the exposure to an additional driver. Bottom line is they would likely be within the law to deny the claim. If they did pay it they would increase the premium or potentially refuse to renew the policy.