How? The parents wouldn't even know.
Sounds like you want to legislate a certain parenting rigor involving a high level of parental involvement.
1) Creating the legislation/framework which is to say put into place the structure about how people can be found complicit and sanctions based on various schemes.
2) Outreach and awareness to the at risk population that help is out there and that the behavior is unacceptable and that something other than a scolding is at risk here.
3) Civil actions in high profile cases which reinforce the message and provide case law to guide other actions.
An analog might be civil actions against bars for serving alcohol to clearly inebriated patrons. We did a booth where we sold beer at an Art & Wine festival and had to get mandatory training on the signs so that we wouldn't do something that made us liable under California's laws if we served drunk people and they later did something bad.
But before we got to the 'how' we should probably make sure that we agree that this a parenting problem. In my day reporting bullying to parents was often met with a shrug rather than action, and school expulsion a temporary set back for bullies.
I'd like to see details of how #1 would actually look. Nuber two is a definitely win. #3 would probably be inevitable because #1 would likely be full of holes and grey areas.
Your analog seems ill equipped to deal with the constant surveillance and responsibility a parent would face. The bartender simply stops serving. End of responsibility. If the patron becomes abusive he gets kicked out by the bouncer. For parents you are talking about a scenario where there is no end of responisibility. You can't just kick them out into someone else's bar.
What kind of punishment are you talking about? If you imprison them what will you do with the kids? If they are poor and you fine them, is that going to be a net win for society?
And teachers? Teachers these days have near zero enforcement options. You can't make them responsible unless you also give them back some power. They sometimes suffer abuse from students as well.
That way nobody ever gets their feelings hurt and no adults have to get off their asses and do the hard work of raising or looking after children, be they parents of bullied kids, parents of bullies, school administrators, teachers, or cops. If we just skunk the internet, we can save all these poor souls from doing any heavy lifting.
Parents wont' have to go to the school or police or other parents about their poor kid. Other parents can continue to say "kids will be kids" and teachers/administrators can continue to watch youtube videos during class while pretending not to be aware of the harassment and violence occurring in their classrooms and hallways five feet away from them.