I personally wouldn't tell a soul, but I'm not you and you're not me. Now I can't tell you what to do, or how to do it, but I can try and put some stuff into perspective.
The greatest animator to ever live was Walt Disney's best friend, Ub Iwerks. He drew anthropomorphisized animals thousands of times a week for the better part of his life (he died while animating). One might say the pair were obsessed with these characters. Yet neither one of them gave a damn what othe people thought. As a result, one of the characters they came up with (an anthropomorphic animal) became the most famous cartoon character in history, earning a star on Hollywood boulevard. Mickey Mouse.
The furry fandom is at its basis, a group of like minded individuals who share a common interest of animals that happen to (sometimes) wear clothes and walk on two feet. That's it.
Now the question is, is it even worth discussing? I mean, it's all so simple. It's not like you're telling them you're gay. Would you be afraid to tell your parents you prefer waffles over pancakes? Hell no. The weird thing is not the interest, it's the shame of the interest. There's nothing to feel odd about...In fact, telling your parents in such a way may make it more odd. Again, waffles or pancakes. Is it worth the stress?