If a parent chargeable with the support of a child dies leaving the child chargeable to the county or leaving the child confined in a state institution to be cared for in whole or in part at the expense of the state, and the parent leaves an estate sufficient for the child’s support, the supervisors of the county or the director of the state department having jurisdiction over the institution may claim provision for the child’s support from the parent’s estate, and for this purpose has the same remedies as a creditor against the estate of the parent and may obtain reimbursement from the successor of the deceased parent to the extent provided in Division 8 (commencing with Section 13000) of the Probate Code.