Pippa Smith said schools should inform parents as well as children about internet safety, so they can make sure its enforced when children are by themselves.
She was speaking after the Internet Matters organisation launched a new campaign to encourage parents to protect their children online.
Internet Matters said half of all parents did not know what their children were looking at on the internet, and the same amount had not installed any form of online content filter.
The organisation warned of several threats to children, including radicalisation, pornography and bullying.
It said 3000 children are bullied online every day whilst around 1600 see a self-harm image.
Pippa Smith told Premier's News Hour: "It's actually total lack of awareness. There will be some [parents] who don't care.
"I know that Ofcom have produced results showing that 15 percent of parents have not even had a conversation with their children about what they do on the internet, so it's a huge problem.
"Most parents I should think probably are on the internet, but I don't they have a clue what their children are getting up to.
"When schools are taking it on board, it should be the parents that the schools are talking to first... Once they're [children] in a situation where they're on their own, all that they've learnt in school will go out the window.
"Parents should always be able to talk to their children, and that their children know that whatever they say that they're not going to be judged, that their parents love them and that they want the best for their children."
Listen to Premier's Des Busteed speaking to Pippa Smith on the News Hour:
