This is not true of Eton (or I guess Harrow or the other public schools).
[0] - Or whatever Boris Johnson got, I can't recall exactly, they were spectacularly subpar but he's masterfully scrubbed them from the internet
So it’s not a very good joke, as it implies that there aren’t any other filters, despite their intense concern for reputation.
The joke would only work for schools that famously looked the other way on pretty egregious stuff if you paid them enough -- not on a school that makes a point of heavily filtering beyond the official tuition. (Sorry, "sticker price".)
Side note: It's also kinda funny how the parent expects 45k to be some kind of jaw-dropping, outrageous number, when Americans routinely pay that much for college.