You can always tell people that you don't want to drink.. without a big emotional conversation.
I mean, if anyone shoves a drink at me, I just say I don't drink anymore. It's not necessarily their business as to why I don't, and I've found people really don't care if I drink or not.
Sure, maybe down the road if you think for some reason that it would benefit your parents in some way to know about your struggles, and after action has been taken, success achieved.. have a sit down, let them know how far you've come from where you once were.. most parents like to know those types of things. But if your motivation is more about reaction/illiciting concern and emotion.. save it. Who does that benefit? Are there any selfish motivations for having some big talk about your issues? You can take care of yourself. Work on you.. enjoy the wedding, take the glass for the toast and set it down. No one will care, the attention shouldn't be paid to you on that day anyways
