If you child is bullying another child, yes parents it is your responsibility. We are guides for our children in this lifetimes thus, we need to check in, know whats happening in our child's life and emotionally be attuned to our children to know when something is "off" or they are not telling us the full story.
What is bullying?
Bullying is when children: tease other children over and over again, ignore other children or leave them out of games or activities, say mean things or call other children names, spread nasty stories about other children
hit and push other children, take other childrens things.
Physical signs: bruises, cuts and scratches, torn clothes
missing property, poor eating or sleeping, bedwetting
complaints about headaches or tummy aches.
Your child might: not want to go to school, stay close to teachers during breaks, start sitting alone, have difficulty asking or answering questions in class, or have trouble with schoolwork or homework, stop taking part in school activities.
Social changes: being excluded at lunch and recess
losing contact with classmates after school, being chosen last for teams and games.
Emotional changes
Your child might seem unusually anxious, nervous, upset, unhappy, down, teary, angry, withdrawn and secretive. These changes might be more obvious at the end of weekends or holidays, when your child has to go back to school.
https://raisingchildren.net.au/school-age/behaviour/bullying/bullying-signs