The twist on @Juno16 's question that always ended up being revealing was “what’s the class that everybody wants to take?” It lead very naturally to how hard to get classes were prioritized, how kids heard about these amazing classes (History of the Holocaust at one school, Yiddish Literature at another, and Creative Writing workshops at another), etc.
Another favorite was how did you meet your friend group? It is a nice opening for the social vibe.
But yes, it should be the kid asking. I saw a lot of eye rolling diring parent questions. A few schools cleverly split tours so students were in one group and parents another, but they were in the minority.