When I first started drawing furry stuff I'd show it to my parents, they thought it was pretty cool, mostly proud that I was practicing my artistic skills. Neither of them really 'got' it, but they were at least cool with it. My dad didn't have anything to say on it, which was a mixed blessing.
Last year, Rainfurrest's charity happened to be the wildlife rescue my mom and sister volunteer at. That was pretty good props for furry.
My dad saw one of those horrendous shows that bashes furry, I don't know if it was CSI or one of the many others. He was freaking out about it for awhile, but he didn't say anything to me, it was just an aside "those people". Just like when he told me the story about the gay guy hitting on him in college, and I reassured him none of my gay friends were going to push themselves on me unwelcome - his response was "Ok, well, I don't know what *their* morals are, but you should be careful."
Other people I'm completely open with, and all my friends are pretty much non-participatory furries - one used to go to conventions but has been out of the loop for a long time, and my mate (who joined tff after I did) is into it and goes to RF with me, but doesn't otherwise participate in the community.
I'm not afraid to share artwork or the fact I'm going to Rainfurrest, but I don't like to really discuss it with my parents because my dad always assumes the worst.