In Washington there is a law that addresses this in the context of a child causing a willful injury to a person or their property.
Action against parent for willful injury to person or property by minor—Monetary limitation—Common law liability preserved.
The parent or parents of any minor child under the age of eighteen years who is living with the parent or parents and who shall willfully or maliciously destroy or deface property, real or personal or mixed, or who shall willfully and maliciously inflict personal injury on another person, shall be liable to the owner of such property or to the person injured in a civil action at law for damages in an amount not to exceed five thousand dollars. This section shall in no way limit the amount of recovery against the parent or parents for their own common law negligence.
What this statute means is that the parents automatically have liability for a child’s acts that can be deemed willful or malicious. This is interesting in that Washington law also holds that a child under the age of 6 cannot be held to have acted in a negligent manner because they lack the mental capacity to have the foreseeability of their actions. So, a child under 6 most likely cannot be deemed to have acted willfully or maliciously if they cannot be said to haven been legally negligent since the standard for mental capacity simply isn’t met. On the other hand, as a child grows in age, so do their opportunities to act negligently and willfully.
Importantly, the statute also caps the parents’ liability at $5,000.00, but leaves open the chances for greater liability based upon common law theories of negligence such as failure to supervise or negligent entrustment (of a motor vehicle, for example).