I feel for you. Your parents are kind of cruel. They shouldn't be so focused on your weight at your age. You are young and probably a beautiful endomorph that hasn't finished filling out yet. You'll probably end up strong and gorgeous when you are older. You'll see.
I used to cycle with a group and every single person in that group was obese except for one or two. I rode with the ones who rode hard and yet I was still obese.
You might have success adding strength training to the cycling. You are probably too young to use weights, but you could do body weight exercise like on Al Kavaldo's website. If you are anything like me, these exercises will be impossible and you'll have to start with his super basic work your way up to the real ones exercises.
Meanwhile, you can eat the diet your parents want you to eat, but if you get hungry, don't reach for biscuits, choose more paleo snacks. Apples with nut butter, hard boiled eggs for example. You don't have to do atkins or a low carb approach, but you can right this minute choose lower carb and more natural foods whenever you do have a choice. So tell your parents you want to have eggs for breakfast and compromise with them by making them hard boiled or poached or scrambled with the minimum of oil. Have them with onions, tomatoes, mushrooms and spinach instead of bacon. For lunch ask for cold chicken rather than a sandwich, and fruit or salad rather than biscuits. Stuff like that.
Female, 5'3", 49, Starting weight: 163lbs. Current weight: 135 (more or less).
Starting squat: 45lbs. Highest squat: 167.5 x 2. Current Deadlift: 210 x 3