I think this just reflects a lot of the insecurities within the fandom.
Basically, if it's not a furry site, then it's not part of our secret club, and people who don't know the secret handshake will get in, which will lead to my mom finding out that I have a fursuit and that I fuck in it... I think that's the logic behind these people. And it's ridiculous. I hope the Weasyl staff doesn't take them too seriously.
I doubt anyone is actually afraid of their parents finding them on the site =P
That makes too little sense, even for furries.
Even though FA is an art site, people obviously use it as a social networking tool. Furries are going to want a social networking site with other furries. There was an attempted furry Facebook (I forget the name) but it failed miserably. Unless something similar comes around people actually use, that's an important deciding factor when people look for a replacement for FA.
If you're looking to make a general art site that's cool. I think that's the main reason people are complaining though, whether they state it or not. As with the ubiquitous logic you spoke, furries don't want their parents finding out they are into X, Y and Z. It's becoming increasingly harder and harder these days to separate your online social circles, creating the necessity for a separate online community than say, Facebook. Furries are going to have a preference for such a social community if it's specific to the subculture.
Also, as someone who is primarily into furry art I know I would want an art site that has furry art on the front page. I can appreciate other art but I want the animal people ;3
Anyway, good luck and I hope it works out!