Does turning off IPhone mobile data prevent phone usage activity on Verizon from tracking my activity?

So, my parents are strict about me staying up late and using my phone and stuff. I don't know why. (Right now I'm on school vacation, and I understand that I should not on school nights). Everyone in my age group does it, and I'm a straight A student. I always get As or A+, not A-. (Class rank #2 btw). I mention this bc my mother says staying up late will make me stupid but I have stayed up late secretly many times in the past, and I'm still doing well in school. I figured out that my dad can see the times I'm using my phone on his Verizon account. He can see if I'm using my phone at 5 PM, 3 AM, 5 AM, etc. So, will turning off my mobile data turn off the tracking? Please respond if you have an answer, and thank you for your time.

That would stop it from logging significant mobile data usage on the Verizon account, but he would still see texts or calls going through. Also if you are at home doing this you should be using Wi-Fi anyway, which he may have a way of tracking too. At some point without Wi-Fi or mobile data I'm not sure how you plan to get online (unless you setup a secret router).