Over half of parents check their child's internet activity at least once a week, according to a survey on bullying carried out by Ámarach.
The poll of over a thousand adults, on behalf of the National Association of Principals and Deputy Principals. also found that 12% of us think cyberbullying is even more serious than traditional bullying.
63% of those surveyed also believe schools should ban smartphones and social networks.
Clive Byrne, Director of the NAPD, said it is up to parents to educate their children to online risks.
Most people believe that parents and schools share responsibility for tackling cyberbullying, the NAPD survey also found, and the NAPD called on the social networking companies to "join the fight against cyberbullying".
Clive Byrne said: "For educators, parents and children, the message is that bullying, whether conducted online or offline, can seriously damage children’s mental health and we must all work together to stamp it out.
"As increased access to technology enables the rise of online bullying, we must renew our resolve to tackle cyber risk, in particular, while at the same time treating bullying, however it presents, as an ongoing serious health risk facing children and broader society.
"Principals tell me that the social networks either do not respond or are slow to react to their requests to take down abusive posts about a student in their care. The social networks ought to have a dedicated liaison officer whose job it is to take calls from schools and parents and act promptly in deleting offensive posts."