Let me just say that over 20 years of doing Open Days and Applicant Visit Days (they have quite different purposes in my discipline as we interview) I have seen a significant increase in "that parent" type behaviour.
At Open Days, staffing my Department's stall, I try to talk directly with potential applicants, and pretty much every parent just cannot seem to stop him/herself butting in. I try to acknowledge their comments and go back to speaking directly with the possible applicant. I can understand shyness, but sometimes I think that with any young person, the presence of a parent (and sometimes both, plus siblings and grandparents, kid you not) changes the way they interact with me.
And then the examples of out & out rudeness I listed earlier. I'm a senior experienced professor - I still volunteer to do Open Days. I do not like being treated as though I'm a snake oil merchant.
I rarely say anything about my title and rank and experience as I just want to be accessible & open and not intimidate potential applicants. It's interesting how when my rank/job (eg HoD or former Dean) is mentioned I get a very different response from parents (mostly fathers, I have to say, but some mothers).
As I say, I feel sorry for the colleagues of some of the parents I encounter.
If you do accompany your DC - and I can see why you'd want to - when they're talking to staff, step back. Even if you know that they're not asking the questions they need to, don't butt in. If necessary go away to a quiet corner, coach your DC to ask the extra questions, and then send your DC back to talk to us again. We don't mind - we want to talk to intending applicants, to answer question. We want applicants who think they'd be a good fit for our programmes. At an Open Day, we're usually there 9:30-16:30. Lots of opportunities to come & chat as many times as the DC needs to. But the applicants, not their parents.