Academics have warned of an "enormous gap" between what parents think their children are doing online and what they are actually doing.
Professor Dafna Lemish, from the Department of Communication at Tel Aviv University, issued the caution after questioning parents and their children about their online activities.
"The data tell us that parents do not know what their kids are doing," said Professor Lemish.
In one part of the study, Lemish surveyed over 500 Jewish and Arab children from a variety of socio-economic backgrounds, asking them if they gave out personal information online.
Almost 75 per cent of the children said that they do, but the parents of the same children believed that only four per cent of their children did so.
The same children were also asked if they had been exposed to pornography while surfing, or if they had made face-to-face contact with strangers that they had met online.
Just over a third of the high school group admitted to meeting with a stranger they had met online. Nearly 40 per cent of these children admitted to speaking with strangers regularly.
Fewer than nine per cent of the parents knew that their children had been meeting with strangers, engaging in what could be viewed as very risky behaviour.
Professor Lemish said that this gap is wider in the US, where children from middle-class backgrounds have more opportunity to surf online privately.
In another part of the study, 30 per cent of children between the ages of nine and 18 delete the search history from their browsers in an attempt to protect their privacy from their parents.
Professor Lemish suggests that common filtering software may not be effective, since children will access what they are looking for at a friend's house, an internet café or school.
And if the child accesses dangerous material outside the home, they will be unprepared and uninformed when it happens, she warned.
Professor Lemish believes that one problem is that parents are not as media-literate as they could be.
"This lack of knowledge on the parents' part may be no different than the situation before the advent of the web," she said.
"Parents don't know what their children are doing on the net, in the same manner that they don't know what goes on at class, parties or clubs."
Professor Lemish advises parents to give their children the tools to be literate internet users, and to navigate around any potential dangers. Most importantly, parents need to talk to their children.
"The child needs similar tools that teach them to be wary of dangers in the park, the mall or wherever. The same rules in the real world apply online as well," she advised.
"For example, under no circumstances should a child ever give strangers their private information over the internet, or meet unsupervised with strangers.
"Children should be encouraged to tell their parents about internet encounters that make them uncomfortable. It's just common sense and parents need to teach them that. Talking with your children regularly is important."
At the same time, parents should not disregard the advantages of the internet. "We tend to forget that it offers our children a source of independence, a way to explore the world, and helps them meet friends whom they could not meet in their real world," she said.
"As parents, we need to help them explore the positive opportunities the internet offers them, and to reduce the risks."
Moon's dark side is mountainous, rugged and never visible from the Earth
The groundwater basins in some areas of Tehran have been damaged irreversibly
This is the first time that any spacecraft on Mars has recorded air vibrations on the planet
Arctic sea ice is thickening at a faster rate during winter, thus slowing down long-term decline: NASA
But, the seasonal ice growth could only delay the demise of the Arctic ice cap for a few more decades