Firstly i would actually get the opinion of a professional if you haven't already - their judgement is an important first step. They will also be best placed to advise you on feelings surrounding shame etc. I always try to think pragmatically in these situations - who really needs to know? For example, if you would like your parents to know regardless of whether or not they're able to support you, then i would state the facts concisely, and explain that you're telling them because you want them to be aware of the reason for some of your behaviours. At least from there they might be understanding.

Do you still live with them? How is your relationship generally? You might need to ask yourself questions like that when considering how to tackle this problem. All the best.