It has been a very long time since I read the book, but unless my memory is playing serious tricks on me, the cathedral was being built by a monastery, not a parish. It would only be a monastery or a diocese that would build a cathedral, not a single parish (a parish is a subdivision of a diocese, and 'cathedral' comes from "seat of a Bishop").

The abbot of a monastery is and was of the same religious level order as a bishop of a diocese, and they were lords of the realm as well back then (I don't know if they still sit in the House of Lords, but they used to). I'm not sure at what level of nobility they were either, but at least equal to a baron and, like a baron, they could "own" or at least govern a town.

Follow-up question regarding the book's social/family constructions. I'm just about to finish Pillars of the Earth on the heels of rereading Connie Willis' Doomsday Book. Obviously, historical novels will lean toward novel as opposed to history, but one major difference between the two involves parental love. I assumed that strong parental love (In PotE, a starving builder says that his children are his best work, though he hopes his cathedral plans will be his second best) was a fairly modern construction, and Doomsday's fond mother who nevertheless prefers to have the nurse mind the children all day was more realistic. How adored were children of poor tradespeople in the 1100s?

PotE is killing me. Wikipedia's list of its historical inaccuracies is fascinating but surely this doesn't begin to touch on the social inaccuracies.