It really does sound like it depends on the school. But here's my experience:
My dad took me to St. Louis and dropped me off at the school, but he took off after that. I wanted him to go elsewhere because I wanted to come off as independent. My dad was fine with that, and he was interested in exploring the surrounding area anyway.
I interviewed the day before Thanksgiving, so I was the only one interviewing that day. At the end, when I mentioned something about finding my dad, the faculty member I had been speaking with was like, "Oh, he could have come along for pizza and the tour," "I like to meet the parents!" ...comments to that effect. I said something about not knowing if it was a good idea for him to come along or not, and he said it wasn't a big deal. It's really fine either way.
As far as my classmates...some brought their parents, some brought their spouses, others went alone. When I see students on interview day, I see TONS of parents.
So, summary. Lots of parents go. They get to see the tour and meet a couple of faculty members. Then they sit around and wait while you go in for the interview alone. It comes down to personal choice. If you don't mind your parents going, I think it's fine to let them go. It wasn't bad to tell them you have an interview. They're proud of you and they care. However, if you think they will just make you more nervous, explain that you might be more nervous and you want to be sure you are seen as independent.