@Joseph
Catholic schools do not turn away students from families in the parish based on ability to pay. They do massive fundraising to meet scholarship needs and operational costs.
Our small town catholic schools have about 35% of the student body receiving reduced tuition (I have friends on the BOD), while large cities of low income/immigrant student bodies (such as Bridgeport) may have 80+ percent of students paying less than full tuition).
Our local Catholic High School has a base tuition of $18,000 per year. Very few students pay 100% of tuition. The high schools have a way to earn extra tuition dollars that is not feasible for yeshivos, They may bring in foreign exchange students under E-1 student visas (you may have seen or read advertisements looking for American Host families). This enrollment may be up to 10% the student body. These foreign students are billed FULL tuition PLUS and additional $10,000 per academic year, plus all expenses: lunch, uniforms, books, lab fees, transportation. It is a cash cow for the schools. (It is also used by MANY non religious private high schools to keep afloat).
These foreign students (the largest group is from China_) are by and large non-Catholics, but must take religion classes and attend Mass, etc. in school, even as known non believers. the schools do not try to convert them, they just want the cash.