I'm an avid anime fan, and I just don't see the same kind of behavior I see in the furry fandom. And there's SO MUCH FREAKIN' PORN in the anime fandom, but I don't see anyone really getting into a huff about it or worry about their family and friends REJECTING THEM FOR A HOBBY OMG.
The problem doesn't stem from the hobby, I beleive, but the fact that furry isn't "out there".
How many parents know of anime, in one form or another? I mean, if they saw pokémon/digimon/beyblade/whathaveyou, they have something to relate to. They have an "idea" of what anime is. It's something that's widely known in society in general.
I don't know for the US in general, but here, in Québec, we've been exposed to japanese animation in the early 80s, even late 70s with many, many shows (most of them considered classics nowadays). So, for most of us, our parents saw us grow with anime...
But when it comes to furry, what is it that parents might have seen or heard of? Most likely, the Sex2K show, or the CSI episode or the Vanity Fair article.
Those hardly represent furry in it's entirety (mostly the more extreme fringes), but they are what most people see as "the norm" as far as furry is concerned. This is what they are exposed to.
This might be the reason behind said "furry closet", much like the "gay closet", exists for various frivolous reasons. Lack of exposure to the "fandom" proper instead of only the fringes of the subculture.
But, as furry gets more "out there", this might change. People will most likely start to see what the fandom mostly is, with the side fringes and core groups, and eventually will end up just like the star trek or star wars fandoms: just another bunch of geeks enjoying something.