The main person legally responsible for the injuries to another is the person (if any) who acted negligently. If the negligent party was a minor, the minor owes the injured party for the damages. Of course, many minors are judgment-proof due to insufficient assets and minors can sometimes avoid debts by declaring bankruptcy, too.
RCW 4.24.190 establishes an action against parents for willful injury to person or property by a minor by saying the following:
"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. [1996 c 35 § 2; 1992 c 205 § 116; 1977 ex.s. c 145 § 1; 1967 ex.s. c 46 § 1; 1961 c 99 § 1.]
Parents are usually not automatically vicariously liable for the torts of their children, but where parents knew or should have known of a child's dangerous proclivities, then the parents may be liable for negligent supervision. Sun Mountain Productions, Inc. v. Pierre, 84 Wn. App. 608, 929 P.2d 494 (1997) (http://www.mrsc.org/wa/courts/index_dtSearch.html). .