Sunflower,
it IS possible that your parents will try to minimize/deny the severity of your problem. It likely will be their own self-protective thing much more than a message 'against' you...that you're too young to know when you have a "real" problem or whatever.
Perhaps you can ask a counselor for help with how to speak with your parents, and what you could say if you do feel that they are heading for denial?
At the end of the day, though, whether or not they can accept your reality it still is your reality -- do what you have to do to achieve your own goals for sobriety...and all other areas of your life, too
Congrats on being so obviously self-aware and wise, and best of luck with your recovery!