We cannot answer whether the parents should be liable in this situation since that is a legal question for an attorney to answer. However, according to the wording in the PAP, the child was certainly a family member and certainly an insured. There is an exclusion on the PAP for using a vehicle without a reasonable belief that that insured is entitled to do so. But that exclusion states that it does not apply to a family member using the named insured's covered auto that is owned by the named insured. This exception to the exclusion is meant to apply in situations such as this since family members, especially children, often use the family car with or without the expressed permission of the named insured.

It may be that the adjuster is relying on a differently-worded PAP, but if the wording on your policy is the same as on the standard PAP, there is coverage for this claim.