According to a poll by BBC Learning, almost one in five children said they had seen something on their devices that had upset them, twice the number parents had thought.
A separate study found that just over 20 percent of parents do not monitor what their children are doing online.
The research was commissioned as part of Safer Internet Day.
While 90 percent of the parents surveyed by the BBC in England said they had spoken to their children about staying safe online when using a tablet or a smartphone, most said they allowed their children to use them unsupervised.
The survey also found that teenagers aged 13-16 were more vulnerable to being bullied online than those aged 8-12. However, parents worried less about the older group using a tablet.
David Emm, senior security researcher at Kaspersky Lab said parents were not often as aware of the dangers of using the internet on tablets and smartphones as they were with PCs.