I heard a good one from another parent on my most recent tour… ‘what’s your favorite class this semester?’ It’s not part of the scripted spiel and you might hear a tip of something for your kid to consider taking (eg, at Lewis and Clark the answer from a couple of students was an intro bio class for nonmajors that fulfills a general ed requirement and I guess they really liked how the material was presented. If my son ends up going there I’ll suggest he at least consider that one for his required science lab course).
As far as whether parents ‘should’ be asking questions… I usually leave space for the kids to ask first but if no one is asking anything (or if I really do have a question I want to know the answer to) I’ll start asking.