Welcome to gyne.org, HeroToXero.
Unfortunately, if you want to do something about your condition, you don't have the option of "ruling out" surgery, because that's the ONLY way to correct gynecomastia. If you've had the condition 3 years and it hasn't resolved, it's not likely that it's going to go away on its own. And as you found, it's impossible to exercise gyne away. Stimulated gland that has grown and become fibrotic won't simply disappear anymore than a girl's would disappear. The problem isn't fat, it's gland, and it's there to stay unless it's removed.
I'm afraid if you want to really do something about your situation, you're going to have to "burden" your parents and you are going to have to accept the idea of surgery. At your age, you've got many years ahead of you where you'll NEED to be able to take off your shirt when the situation calls for it. There's no sense in being unhappy and embarrassed any longer. Get the surgery and be happy. The good news is that it's not a bad surgery at all and the recovery period is relatively quick. Summer would be a GREAT time to get it taken care of while you're out of school.
Don't worry about how your parents will react. It's nothing you did and it's not your fault. The condition is actually very common. Your parents wouldn't want you unhappy and embarrassed anymore than you do. So trust that they love you and will try to help, if they can. Surgery is typically in the $5,000 to $8,000 range. If they don't have the money, they would need to look into getting a loan and making it a priority, just as they would prioritize getting a car or getting a leaky roof fixed. Surgery, in my opinion, for a young man with gynecomastia, is not a luxury, it's a necessity. Since it interferes with the ability to live life normally, it simply must be addressed, and the sooner the better.