Expect slow posting to continue indefinitely. Meanwhile, let’s continue with DMC’s look at the role of the believing community, which I think can be taken broadly in this context:
11. The Christian communities to which the individual families belong or in which the children live also have a responsibility toward children baptized in the Church. By giving witness to the Gospel, living communal charity, and actively celebrating the mysteries o Christ, the Christian community is an excellent school of Christian and liturgical formation for the children who live in it.
Within the Christian community, godparents or other persons noted for their dedicated service can, out of apostolic zeal, contribute greatly to the necessary catechesis in the case of families that fail in their obligation toward the children’s Christian upbringing.
Preschool programs, Catholic schools, and various kinds of associations for children serve these same ends in a special way.
By my comment above, I would certainly include parishes. But I think Catholic schools also should exemplify these values, not leaving them to the parish, nor focusing too much on didactic education. Thoughts?
Catholic Sensibility 
Look at what the Diocese of Rochester just did?
Yes; well, the diocese of Rochester has issues. Their schools have become somewhat detached from parishes; they have regional student bodies and regional organization. It has disadvantages in losing connections with parishes–the important touchstones between faith, Sunday Mass, intergenerational support and other similar possibilities.