Do you have any other symptoms of thryoid deficiency? (lethargy, fatigue, dry skin, constant thirst, hair loss etc.?) I have been on thryoid meds for a few yrs and I was gaining weight too and never had any issues with weight in my life. It's worth checking, but it might be unusual at your age....I only say that because you say you don't eat much and gained 1.5" in 5 mos.
Your parents definitely sound controlling an manipulative. I am familiar with that. I wasn't equipped to move out at your age either, but if you can find a way, you might want to do that. Call their bluff, if you can swing it. They will still support you if you were in dire straights. You are their daughter. They will always love you (despite acting they way they do). A control freak isn't just going to let go that easy. I'm twice your age and my dad still tries to manipulate me and offer me unwanted help/advice. Are you an only child?
You could find faults with them and give them a taste of their own medicine and see how they react if you feel devilish.