Not sure how strict your folks are, but all of the problems you describe (phone heating up, quick battery drain, etc) are all signs of spy programs installed on the phone. The number 1 way to tell is if you get a new battery and the problem persists. Flexispy among other programs will cause this because the bridge that allows others to read your texts, emails, listen to phone calls and get your GPS locations whenever they want, all use data. When your phone is idle, watch your transmission indicator. If it comes on randomly, especially after you rcv texts, emails, etc, then your phone could be sending info to someone else, like your parents, or spyware may have been installed by a program you downloaded.
Hope that's helpful and doesn't just make you paranoid.