Can you find out what you do on your iPhone?

Okay that question was kind of stupid, lol. But I mean my parents got me an iphone 5 and they told me that they can call the cell phone company and find out everything I do on there. Like my search history, apps, and things like that. We have Verison by the way. I don't do anything bad on there, but there's just some embarrassing things because of my friends haha.

Anyways, can they actually do that or are they just saying that so I don't do anything bad? I just don't understand why they don't trust me and it bothers me.

Well, actually, yeah, you can see all your history, even if you've deleted it (well, I guess all internet history up to a certain point). And, you can see all your apps, even the ones you've deleted.