Agile's main innovation is organizing projects as a (mostly) always shippable series of iterations (i.e. "what do you want next"). But the actual "how" with cards, points, sprints, boards, workstreams leaves a lot to be desired if parallelism is the goal. Communication costs are really high when every one is micro-siloed, hand-offs, which are serial, are costly and involve a lot of relearning the same context, and there is more integration work which is also serial.

I think it's possible to have good parallelism within Agile but I don't think it's the Agile that makes it happen.