Despite this, only 9 per cent of parents in the survey had installed any filtering software.
And though many parents did choose what their children under 8 accessed online, 59 per cent of youngsters over the age of 8 decided what they wanted to look at themselves and 72 per cent reported to be using the internet without any parental supervision.
The researchers talked to more than 700 children aged 6-14 years old and their parents. Of the children, 88 per cent watched TV, 66 per cent used the internet and 36 per cent listened to radio daily.
Seventy-two per cent of the children also had access to a tablet and 48 per cent had access to a smartphone.
The report highlighted the difference between ethnic groups, with Maori and Pacific children less likely to use a tablet, PC or laptop than their Asian and European counterparts.
NetSafe's chief technical officer, Sean Lyons, said that as internet access increased, keeping children safe could be challenging.
However, he said, even software filters to stop access to unsavoury sites are not foolproof. "You can't just flick a switch, walk away, wipe your brow and say, 'Whew ... it's now safe'."
Mr Lyons said it was better to prepare children to deal with online dangers. "The challenge exists, but how you handle the challenge can impact how it affects you.
"So we need to know what the children know and know they know how to deal with such challenges."
The Parenting Place's writer and researcher, John Cowan, said parents were worried about "what mischief their children can get up to and what mischief can be done to their children".
He said making use of the parental controls on the differing devices was a good starting point but building "good character" in your child is the best form of defence.
He said while filtering software is improving, any "motivated, intelligent child" can bypass such security.
"The real security filters are those that you install in their head, rather than through a machine," he said.
Despite these growing security concerns, the Colmar Brunton survey showed parents were still making good strides towards managing their children's use of technology.
Almost half reported putting restrictions on their children's use of television and the internet.
Broadcasting Standards Authority chief executive Karen Scott-Howman said it was great to see parents helping their children navigate their way through the media landscape.
Children themselves also had a greater awareness of the online dangers and unsuitable content, with close to half switching channels or closing their browsers if they didn't like what they saw.
Safety tips for parents
•Talk to your children to ensure they understand the risks of using the internet.
•Go online with your children and educate them about what is appropriate and what isn't.
•Install site-filtering software and regularly check it's still working and hasn't been switched off.
•Monitor the amount of time children spend online.
•Set up parental controls to restrict access to certain sites on your children's mobile phones and tablets and explain to them why you've done it
Read the full report online here:
