I think that it is probably a combination/mix of parents. Some kids push themselves, some kids just ARE and others have the potential and the parents push them to do it.
as for mensa - well it doesn't bother me what other parents do there. I think for some people it is important and others it isn't. I sat a mensa test (I asked to) when I was 16 because I couldn't understand why I couldn't do well in exams and started to think I was really thick. It boosted my confidence no end to feel I was good at something. I didn't join mensa though, it was just something I wanted to do for some reason.
Would I get my children to do one - no. However if they expressed an interest in the types of puzzles or wanted to do something when they are older then that is up to them and I would support them. I think for some families it might be a way of trying to work out how to challenge their children, for others it is a boasting thing. Who know.
Taking a mensa test though won't harm a child in any way so it doesn't worry me if people want to do it.
I do find things like this interesting though.