When parents discover Twitter, teens get lost

Will students put money in Twitter?

Teens may love Twitter now, but the minute their mothers sign up, they're taking the conversation offline. And that should have Twitter worried.

Teens are a key demographic for social networks because they are an index for "cool" and can influence virality, said Amish Jani, managing director at FirstMark Capital.

(Read more: What companies are doing with your intimate social data )

"You can build a very successful service without teens [e.g., LinkedIn], but they can be a leading indicator for headroom in new-user acquisition," Jani said.

Twitter risks losing its edge, however, if it makes the same mistakes Facebook did, some teenagers say.

(Read more: With teens in mind, Facebook pushes both ways on privacy )

Students in the investment club at Warwick Valley High School in Warwick, N.Y., told CNBC they would stop using Twitter for two primary reasons: One, if Mom and Dad start poking around on the social network; and two, if annoying ads start appearing in their feeds.

For teens, social media is all about saying whatever they want to whomever they want, and they're not about to let the rents crash their party. The minute parents join, the fun is over, the students said.

"We [high school students] went from MySpace to Facebook to Twitter. I think when the parents really start coming into Twitter, I don't see myself being on it as much," said D.J. Mendenhall, a senior at Warwick Valley.

And if you're thinking teens can just block their parents, think again.

Dean Belcher | Stone | Getty Images

Most said when they tried blocking their parents on Facebook, they got in trouble. Instead, they limit their Facebook use and rely heavily on Instagram, Snapchat and Twitter as their go-to social media platforms.

Teens clearly aren't fans of ads on Twitter.

While current ads on the platform can generally be ignored, the company risks young people fleeing the network if obnoxious ones start appearing, the teens said. The students added that they'll be forced to go to "the next big thing in social media." (I think that was a threat, Twitter.)

But advertising on social media is obviously seen as a big opportunity.

"We believe Twitter will be a key beneficiary of the large-scale trends toward social advertising and mobile devices, and therefore see significant growth potential and operating leverage in Twitter's model," Cantor Fitzgerald analyst Youssef Squali said to clients in a note.

However, he wrote, Twitter's growth expectations could be affected negatively if the platform fails to attract big brand advertising budgets.

By CNBC's Seema Mody. Follow her on Twitter @SeemaCNBC.