Yes its bad. You don't need to loose weight and as a teenager you should be eating 2000+ calories a day to maintain your weight. Probably 2500+ with all the exercise you do. Try uping your calories by 200 a day till you get up to that amount, and ask your parents if you can get some profesional help.
It is indeed bad to eat just 200-800 calories a day. I don't know how tall you are but I would say if you are over 5'2", your goal of 110 is unhealthy. I am 5'2" and small, and I don't weigh 110 lb.
You are still young. Your body will continue to evolve. I strongly recommend that you listen to your parents. Perhaps ask them to get a nutritionist for you. You need to find out why you work out all the time and not losing weight. Perhaps you are eating the right kind of food.
Screw those those jerks who call you names. Really, who the heck cares what they say. This sounds like just a cliche but your inner beauty is a lot more important than what you see outside. What you see outside is really nothing. They come & they go.
Good luck with you and stay beautiful.