One in three teenagers become victims of cyberbullying, which works out to be double the amount recorded at this time last year, according to internet security vendors McAfee.
McAfee polled 2,000 children aged 11-17 and found that 35% had been bullied online – up from 16% 12 months ago.
Some 40% of those questioned said that they had witnessed some sort of cyberbullying, which is also a doubling of last year's results.
In addition, thousands of teenagers under the age of 15 are using picture messaging service Snapchat and dating app Tinder, which parents have unwittingly helped them set up.
However, the report adds that only 27% of parents are concerned that their children may be subject to bullying online, well down from the 47% last year. But 67% of children are not allowed to use the internet without supervision, the report adds.
Slightly over three quarters of the parents that were polled said that they have spoken to their children about the perils of the internet.
"The responses from McAfee's survey shows that there is a real gap between parental concern and the reality of what children face online," said Andy Phippen, professor of social responsibility in IT at Plymouth University.
"While it is encouraging to see that these conversations are happening, there are areas in which parents may not be completely aware of their children's online behaviour.
"It's now time for parents to take the conversations to the next level and become further educated on the social platforms that exist, what ages they are suited for and what type of behaviour they encourage.
"Cyberbullying happens across all platforms and children's use of social media is transient."