Not only do they pay for it, they pay for public schools too.
I am, by any measure, at least upper-middle-class (though I do have to work for money; it's not free in my mailbox). I don't have kids, so I wouldn't be taking advantage of the public schools I pay for, but I also grew up in a very median-income household and went to private schools (which were much less expensive at the time, though still not cheap). So my parents paid for public schools that neither of their children ever attended.
I am, by any measure, at least upper-middle-class (though I do have to work for money; it's not free in my mailbox). I don't have kids, so I wouldn't be taking advantage of the public schools I pay for, but I also grew up in a very median-income household and went to private schools (which were much less expensive at the time, though still not cheap). So my parents paid for public schools that neither of their children ever attended.