I think this is incorrect on two accounts: Yes, transformers and individual layers are parallel, but the entire network is not. On a first level, it's obviously sequential over generated tokens - but even generation of a single token is sequential in the number of layers that the information travels through.
Both those constraints are comparable to the way humans think I believe. (The human brain doesn't have neatly organized layers, but it does have "pathways" where certain brain regions project into other brain regions)