Not criminally, unless the parent in some way facilitated or assisted the murder; if the parent were involved in the murder, he or she could be liable as an accomplice (also called aiding & abetting) or as a  conspirator. If the parent helped hide the child from the police or dispose of evidence, etc., then the parent could be liable for such crimes as interfering with an investigation, destruction of evidence, etc.The parent could be civilly liable (that is, could be sued by the deceased's family) if the parent were negligent, or careless, such as by not warning the authorities of a credible risk or threat of violence, making weapons available to a minor, not making sure that a minor child with mental health  issues took his/her medication, etc.