Really, the "Am I a furry" thing boils down to "Do I consider myself a furry?". One thing you've got to remember about stereotypes is that while there was/is SOME element of truth in it for at least one member of a community, is that the loudest are usually the most heard, and stereotypes don't apply to everyone.
About your parents, I just told mine "it's essentially like any other fandom or subculture, Star Trek, Anime, it has its bits, it has its people with a different level of involvement". Now they're fine with me commissioning a partial fursuit and a collar with a bell attachment! (Well, my mum is)