On the other hand, certain people like to say that quantum computing is not about simply computing every possible result of something in parallel universes, that you can't do that much better than with classical computing. If that's so, doesn't it imply in some vague way that any parallel dimension is something less than completely real?

Perhaps it's consistent with computation being done in many "parallel universes," but with heavily restricted communication - in this model it's almost trivial to parallelize work (flip a quantum coin, do A if heads else B), what's tough is making sure that you don't get parallelized along with the work (i.e. that you don't become entangled with the computation) and making sure there's a way to accumulate the parallel computation into something that's singly-readable and useful.