Although in somewhat of a less extreme way, I was actually thinking the same thing.
By going through her parents I face the possibility of getting a "No" on more than one front. By just asking her, if she says "I'd really like to, but I'll have to ask my parents" it for one, saves me some trouble, and two, if she's willing to try and convince her parents to let her go out with me, that shows how interested she actually is.
Of course, that also leaves me with the possibility that she could claim her parents said no, when in fact she never asked and is just looking for an excuse not to go out with me. Although I consider that possibility extremely slim, it's a possibility nontheless.
I dunno, maybe I'm just overthinking this whole thing now. Maybe it's as simple as just asking her out like I would any other girl.