But that's the fine line that parents must walk -- establishing rules but maintaining trust and respecting their kids' privacy, Shapiro said.

Over the last year, several phenomena as well as government interest have brought Internet safety into focus. Cyberbullying incidents, so-called "sexting" or the sending of explicit images or messages, the loss of jobs over photos posted on social networks have all propelled parents to educate their kids, Merritt said.

Parents often underestimate the amount of time of time children spend online. Kids in the U.K. said they actually spent 43.5 hours per month online, far more than the 18.8 hours their parents thought.

Some children also maintained that their parents often do not know what they're looking at online. Eighty-six percent of parents surveyed in Australia felt they always knew what their kids were looking at online, but only 65 percent of Australian youths agreed. That gap was the largest among the 12 countries surveyed.

The survey did show a level of agreement about how text and instant messages and other short forms of contact are contributing to a decline in language skills. Sixty-three percent of parents felt online messaging makes it harder for kids to learn to write well; half of kids agreed.