Purely a hypothetical, of course!- but if the option was there to make an up-front annual payment to your DC's school to 'improve' things like staff:pupil ratios, or greater small group learning opportunities and so forth, would you?
What about a means-testing of contribution? Please bear in mind this is a hypothetical! I am maybe imagining a Free School of the future which has legally been allowed to fly the governmental clutches in order to do this, say.
I know where I live I believe the reality is most parents could afford £1000, so if there are 60 DC in a year, and say 2/3 of parents pay that sum (or all pay less, whatever) you'd have enough cash to be able to reduce the class sizes to 3 of twenty DC, if not full-time, then much of it!
The only 'rule' I would like to impose is that before you explode at the unfairness and iniquity of this as not all parents can spare any money and that this can only benefit 'the wealthy' I want to ask how vocally you currently oppose private education (where he who pays wins) and grammar schools as they only benefit 'the clever'. Just a thought.
This has gone through my mind as I pay just about that per year to get DS2 tutored as it is!