When a parent is required to pay for damages caused by their children it is an example of Vicarious Liability.
Vicarious Liability usually occurs in cases where a person such as a child caused damage to something that belonged to someone else and is unable to pay such as being a minor or having financial instability.
When a child breaks or damages something the parents are always responsible to pay for the damages.
When a parent is required to pay for damages caused by his or her children it's also called civil parental liability.
The reason it's called civil parental liability is because it's non criminal and the parent is only obligated to financially compensate the party harmed by his or her child's actions.
Parental civil liability is intended to compensate victims of torts (wrongful acts) and to encourage parental control and supervision.
Parental civil liability generally begins when the minor is between eight and ten years old and ends at the age of majority, which is between 18 and 21, depending on state law.
Almost every state has some sort of parental responsibility law that holds parents or legal guardians responsible for property damage, personal injury, theft, shoplifting, and/or vandalism resulting from intentional or willful acts of their un-emancipated children.