You're 16 years old, that means you haven't stopped growing.
When I was 16 years old I was 5'3, now I'm 20 and I am approximately 5'6, a year and a half ago I was 5'5. You see what I'm saying?
Your father is right, go to the gym, play basketball, do rope jumps and don't forget nutrition. I wish my father would have known this.
Ironically he just entered the room right now and asked me to stop staring at the computer but go out and train. Yeah right, now he says that
That plus will power will probably give some more inches.
My guess is, since you haven't stopped growing you can still gain a couple more inches.
LL is not for you. Not at this age. Better wait until you're 21-22. That fact that you're wondering about what your parents will think just shows that you're not mature enough for LL ( not that I am). LL is something that you should decide on your own. It's your decision, your body and your life.
Learn as much you can about it, ask a lot of doctors, get a lot of opinions but in the end it's your call.
GROWtalORdie is damn right, you have to do it for your own purposes, not for a girl or boy.