Not to pick a fight with Hawking, but isn't the existence of parallel universes an extra assumption above and beyond the math? The math doesn't say anything about additional universes, just what the result of measurements is. In fact, the multi-universe interpretations of the math were developed to match the theory, so the fact that theory (or experiment) agrees with them isn't really evidence that they're correct, is it?

This is why, I thought, the Copenhagen interpretation was the most widely accepted, while the others introduce extra complexity.

(I say this, by the way, as someone who prefers the many-worlds interpretation...)