There are a few ways it can be proven.
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If the parallel universes are different versions of our own, I can use them to get information I couldn't get in my own universe. For example, if it's a universe where North Korea has an open border policy, I could walk in and observe my surroundings, then end the dream and describe things about North Korea nobody knows about. Though this is not super hard proof.
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I could venture into the parallel universe to get a better understanding of physics, and people in my universe would know something is up because I came back knowing about how things work that the average person could never know. Though, again, this is not super hard proof.
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I can play Marco Polo with other sleepers from my universe to see if I can find them in the parallel universe, using a system of identification that cannot be impersonated if any entity tried. This would be hard proof, but it would require that people end up in the same parallel universes.