Ahhhhhh I know how you feel dude...
My mom disaproves of everything I do, and giving the way she treats/talks about my freinds who cosplay, most likely covering it with fur would not make it better
But on the other hand, although she hates my music, she likes my writing and drawing soooo
Pretty unpredictablexD
I'm tempted to tell her that I'm a furry so I can go to furnal equinox this weekend, so if I do I'll let you know!
As for you
Your parents don't sound as evil as mine, I'd either not talk to them about it, or just explain it like "I am what I am, an thats an anthro-artist, it's not illegal, it's not weird, it's just different and my way of being myself."
Hope this helps