There are 2 different things:
Parsonage which means the school pays rent/mortgage and other living expenses, utility bills etc. People who get Parsonage actually have to pay full social security taxes on it and it’s on their tax return. It is not counted for benefits etc but it’s on their taxes.
School checks (it had an official name) is where schools can give money to other schools to pay their staff’s children’s tuitions. It’s iffy and not all schools will do it. That money is tax free. And not counted as income
