I never really had that problem, nor will I have to worry about that happening. My mother was the one that pointed it out to me in the first place when I was browsing this fursuiter's page on DA and she really liked it. I guess she was the reason I found this whole community. She even said that she was interested in it, but she says it would be weird for her to become a fur since she isn't a "young person".
One thing you could do is break it to them gently. Stage it, in a way. Like just pretend to browse the Internet and pull up a page talking about furries. Bring it up to your parents and see how they react in the first place to furries. And if they react positively to it, you can tell them you are a furry. If they react negatively, try dropping subtle hints. Coax them gently into being fond of it.
I may not be the most helpful person but hopefully some of that was useful.