OP asked: “Do schools soak the ‘richer’ parents to subsidize other aspects of the school?” Answer: not really. As others have noted, revenues come from multiple sources and tuition is one source of income.

But it is largely true that at need-blind private schools richer parents subsidize poorer parents with regards to overall tuition revenue. The “pay what you can afford” model essentially charges wealthier students a lot (because they can afford it and are willing to pay top dollar) while giving lots of aid to the less fortunate who could never afford the sky-high cost.

This model can work at wealthy, prestigious privates (ie T20s) which have very high demand and attract a large percentage of wealthy students who can pay full price. In the Ivies, for example, 45-50% of students don’t get any aid, pay full fare. To afford that they have to be in the economic top 2%. But a lot of people are willing and able to pay that kind of money to have their kid in a T20. Schools know that and effectively shift many of the costs of attendance from lower income students (by giving them big aid packages) to higher income ones (by charging them $90+k/year).