First of all, a simple question.
Are you sure you wanna tell your parents about it ? A lot of people (and I'm one of them) never tell their family about the fandom, and this way nobody gets hurt.
Think. If you tell your parents, you have a 99% chances they will google it ; and we all know that the internet isn't always nice with us. That would lead your parents to believing the wrong thing. And that would suck.
So if you believe it's necessary to tell your family, leave no place to doubt about what a furry is. Really. No place at all.