>is there a parallel usage which goes the opposite way, putting down the Oxbridge achievement?

No, but these days "redbrick" is not necessarily a put-down. It's a bit like "brown stone" in the USA. People are rather proud of having been to a "red-brick" like Manchester University.

To find a put-down you have to look at the word "breeze-block" used for those, post 1960s Universities (Sussex, Salford, UMIST) and former Polytechnics which didn't even get even built of bricks!