Former Admissions Director here. I expected both parents to come to the interview. If something conflicted with that, I expected to have been able to at least chat with both parents at some point earlier in the process. Because I made it clear that both parents were expected at the interview, usually both appeared. The exception of course was if they were divorced/split and one parent was not in the picture at all. If there was another adult who played a major role in the child's upbringing such as a step=parent, that person was welcome to come. My reason was that I wanted to make sure both parents were on board with the program, supporting their child there, the tuition, etc.
We did keep track of attendees to the Open House as we had them sign in on a computer and for weekday tours they registered. I would have expected parents to attend at least one tour in the year before applying, ideally both parents, to make sure they understood the program and there would be no surprises. However, if it were just one parent who took the lead that was fine, as long as both parents were in evidence at some point.
Part of success at this particular school was parental support of the program and the student so making sure that was in place was very important. It wasn't just for show; we truly wanted to make sure that support network was in place.
Scheduling a private tour is a PITA move, definitely, but if you're polite, give a choice of dates, and explain that every other event just doesn't work somehow, it would be OK. Best to try to make it to the scheduled events, however. Most schools try to plan several and at varying times in terms of weekdays and weekends.