Visit the school with your parents at some time independent of the ASW if they're all dead set on checking out apartments and restaurants, etc.. There is no benefit whatsoever for them to attend the ASW -- it's not called Admitted Students and Family Weekend, after all.
Their presence at ASW will interfere with you getting to know potential fellow classmates and the school generally because you're going to be the only one (or at most one of a few out of hundreds) w/ your parents there and will thus be perpetually red-faced and anxious.
The ASW invitations I've received so far (Northwestern, Penn) go out of their way to make clear that partners/spouses are welcome. None say anything at all about parents -- which makes sense because this is professional school and not undergrad. Do your parents insist on chaperoning you on dates? Do they still take you to the doctor?
Granted, I personally wouldn't really even need to attend the ASW to determine whether *I'd* go to Columbia or not (I'm on hold and would jump at the chance), and I'm sure you won't be forever branded with shame b/c your parents tagged along, but it's just really not appropriate. I know lots of law and med students who formed friendships that ended up being the core of their social group at school during ASWs. Being the *kid* ushered around by his/her parents is not going to help with that.
You're an adult and your parents, while proud, really don't have any true role in the process. This is your decision and they should respect that and keep their distance. It's for the best.
I mean, I'm 99% sure I'm going to end up at Columbia. I'm attending the ASW because I've never seen the campus and want to look at the apartments before I fill out my housing application.
My parents usually aren't overly involved in my life, but they are with the law school thing. It's because I'm their first born and Columbia was my dream law school so they're all proud and stuff. I guess I'll try to explain this to them. Thanks for being honest (seriously).
...I think I might tell them I can't come home because it's supposed to snow and attend the other ASW at the end of March alone, because I really don't feel like being that