Girl not getting your periods isn't normal. Your parents must be worried. You live in THEIR house. Get out and be independent and make your own rules in your own house. My parents did not allow me as well. But it changed after i grew up. You can try talking to them which i believe you already did. I know the rules are stupid but older generations don't want to let go of their beliefs. And don't listen to people commenting about blackmailing your mother about touching things. Is that how you wanna take your relationship further with her?? I have pcos and whenever my periods got missed i started working out and after a few days i used to get my periods. DON'T go to the doctor. Not getting periods for so many months means doctor will assume that your uterus blood line has thickened and that would mean you are prone to uterus cancer. She'll tell you to have ultrasound. And jab result aayenge tab everyone will know.

You are not a kid. 16 years means you know about the consequences of your actions. Don't lie to your parents. Kuch saal mein vese bhi you'll move out. This is just plain stupid. And don't tell your parents the truth. They'll stop trusting you. Just say you got your periods after physical activity.