Can my parents see what apps I am using on my phone?

[jtyoung0826]

I have an android phone that is hooked up to MY gmail address and is on Verizon and well, my question is...is there any way for my parents to see what apps I am using on my phone without actually HAVING my phone in their hands? I know that, if they had my email address and password they can see it. But is there any way for them to see what is using the internet on my phone? For example, is there any way for them to be like "oh he used the Twitter app to connect to the internet on December 6th!"
Now, I'm not talking about "can they see what apps I've bought" because I'm not gonna PURCHASE any apps...and even if I did purchase them, it'd be directly from MY credit card.

And one more follow up question...

If I have my phone on airplane mode, and connected to Wi-Fi...there shouldn't be a way for them to see my internet/app usage then...is there?