I don't think you need to be Pierre Bordieu to understand that cars, like any object, are not just a 'thing' but convey cultural and social meaning. They signify you membership of a tribe....or don't, that's in the eye of the beholder. Those with more economic capital have more choice in how they convey that message, because they have more choices in terms of how (much) they spend their money. Oh, and expensive things are (on the whole) nicer than cheap things.
I guess parents at posh private schools want to show that they fit into their tribe as much as anyone else, maybe even a little more so 
. My daughter goes to one of these schools and the Disco's are all being part-exchanged it seems. Nowadays, it's all about the Volvo XC's and the Mitsubishi Plug-In. I blame Greta Thurnberg.
I've just bought a Mini. I'd love to say it is because it's the best value and best performing car for my needs. Of course it's not really. A Toyota would be just as good, but I'm a fortysomething, middle class, lefty-liberal professional parent, and that's what women like me drive (not all of them, and probably not the cool ones, mostly the ones trying to prove something, I would have thought....)
Finally, I've put all those hours listening to Thinking Allowed with Laurie Taylor to some use.....