More than three quarters believed their pupils' parents relied too heavily on schools to prepare kids for the online world, and one in three felt parents didn't know enough about cyberbullying.
Featherston School principal Phil Robertson said cyberbullying wasn't prevalent at school, but some children were experiencing it at home.
"We don't allow kids to use their cellphones at school. It's probably not an issue for us yet, but it's certainly something we're aware of."
Primary school-aged children weren't old enough to be on Facebook, but Mr Robertson had been forced to warn pupils to be aware that, while it wasn't the school's "business", they needed to consider how public their comments were.
"I've had a couple of parents in the past concerned about what they were seeing on their daughter's [social media] sites when they check them."
Teachers were well equipped to teach students about being responsible and respectful, which covered online behaviour, he said.
According to the survey, 77 per cent of teachers thought internet safety should be a dedicated part of the curriculum.
After cyberbullying, the next biggest issue teachers had was students coming to them about viewing inappropriate content online.
NetSafe executive director Martin Cocker said there was room for increased training and resources for teachers in terms of dealing with online incidents and cyberbullying.
"There is a greater need for teachers to feel confident dealing with these issues, and feeling they have access to the tools to deal with it and manage with the challenges students face online."
APNZ, additional reporting Nikki Papatsoumas