How to convince parents that a breast reduction is medically necessary? (16 year old)

After sitting down with your parents and explaining your side of the issue. You cannot legally proceed with a breast reduction unless your parents sign the consent. I suggest you sit down with both parents again and explain functional reasons why you want a breast reduction.

If you are hoping to have the surgery on health insurance, rather than paying, don't get into arguments with your parents because it will not be covered until you turn 18.  In the meantime, as Dr Pousti explained in detail, you should begin to take the steps required for precertification. This is a lengthy process, often requiring conservative medical treatment and documentation which will take up to a year, so you will be doing something directed toward your goal; physical therapy may alleviate your pain pattern in the meantime as well. Best wishes.

You must read this blog which is essentially a 17yr old young woman's story about her feelings, her motivations and her parents attitude.  It also talks about her experience before and after surgery.  It should be very helpful in your conversations with your parents.  good luck.

I'm sorry to hear about the physical and psychosocial problems your disproportionately large breasts are causing you. 
It is probable that you are dealing with juvenile breast hypertrophy along with the physical and psychosocial consequences of this diagnosis. In other words, the breasts are too large for your frame causing  both physical and psychological distress.
In my practice, timing of breast reduction surgery is determined on a case-by-case basis. It is NOT too early to do your homework and learn as much as possible  about  breast reduction surgery and the potential risks and complications associated with  the procedure.   
As you learn about breast reduction surgery, make sure you read about the potential risks/complications associated with the procedure. You may find the attached link helpful in this regard.  For example, unsatisfactory scarring is  one of the potential complications. Make sure you also understands that further surgery may be necessary in the future (for example if the breasts were to grow in size again).
On the other hand, breast reduction surgery is one of the most patient pleasing operations we perform and I think that for the right teenager (enough symptoms) it may be an excellent option (regardless of the age). I agree with you that having your parents understanding and consent will be  important;  I would suggest that you sit down and have a calm and descriptive conversation of the types of issues your disproportionately large breasts cause for you. You may wish to show her the physical signs of breast hypertrophy, such as bra strap grooving, skin irritation…It may also be helpful if she reads informative websites such as this one and/or joins you on consultation with well experienced board-certified plastic surgeons.  Talking to other young ladies who have undergone this procedure may be helpful as well. Sometimes breast  reduction surgery is covered through health insurance. The best way to obtain insurance coverage for breast reduction surgery involves some “hoops” to jump through. The more documentation you have (for example, from your primary care doctor, physical therapist, chiropractor etc.) the better when it comes to obtaining insurance “authorization” for the procedure.This documentation and letter/pictures from your plastic surgeon will help you obtain authorization.
 When the time is right,  seek consultation with well-trained/experienced board-certified plastic surgeons.

I hope this, and the attached link, helps.