Since it was technically the dog that discovered the diaper not your parent, I wouldn't "square up" with them and cause a big fight over this. If you had a stash in your room, not in plain view and they went out of there way to confront you about it, that would be a different story. Like many people have said, you are over the age of 18 and deserve full privacy and respect just like anybody else. Parents always forget this no matter how old you are.
If you were aware, that it was in the trash, and knew your dog is snoopy and chews things up, well, you had it coming. If you had a garbage bag you could just keep all the used ones in there. Once your parents aren't home, go dispose elsewhere, not at home. Ever. To remain anonymous from everyone. If that's what you want, you have to really take extra precautions and make sure your never going to end up in a situation that isn't explainable.
What I mean by this, is; even if you do get "caught" its your parents (or whoever "busts" you) that is in the wrong, not you. Like I said earlier, a parent going out of there way, snooping in your stuff, that is concealed, in your room/car or not in plain sight.