Yes, because if a 14-year-old child has a personality problem, it is because his parents didn't pay enough attention to his needs and feelings. Why should any child, in this case a teenager, wish to harm other people? Obviously, he had some personality problem. On the other side, the parents should have had that gun kept in a secure place, out of his reach.
The parents of the 14-year-old gunman that wounded three at a South Carolina elementary school should face some consequences. Particularly, the parents should be held accountable for the teen getting access to a firearm. If this firearm came from the home, the parents should face liability for the shooting. Also, the parents should be investigated to see what else was going on inside the home.
Parents might not always be in a position to prevent their child from doing something bad. If a parent has joint custody, they might be begging the court to prevent the other parent from allowing the child to drink alcohol, do drugs, or spend time with poor influences. Most of the time, the courts don't care. It's not the parent's fault if they've asked a court to intervene, the court says no, and the child then does something bad.
No, the parents should not be held responsible for the 14-year-old gunman's shooting of three students at the S.C. elementary school. Students are always taught at school to be responsible for their own actions. Parents are often encouraged to help teach students personal responsibility. This would be another case. If it's found that parents didn't do what they should, such as making sure the student got to school, etc then hold them responsible for that.