As they were parochial schools (i.e. part of the parish) the pastor was ultimately responsible for them and typically part of their finances came from the parish collection plate (I've heard that there were some schools with no tuition, they were fully funded by the parish). Daily administration was left to the religious order which staffed the school, the principal was often also the mother superior of the convent, but the pastor would be the one on the hot seat if something went wrong, so they were probable paying attention. The same order which staffed the school I attended (1958-1966) also operated the local Catholic hospital. The pastor would go to the individual classrooms to hand out report cards at the end of each semester. We would have a weekly school Mass (usually said by the associate). All of the parish's altar servers came from the school and would be pulled out of class for funeral Masses.