Deutsch, The Fabric of Reality, Chapter 2, experiment
I finished chapter 2 of Deutsch's The Fabric of Reality and I think I can describe an experiment that could be constructed to test whether there are parallel worlds or not.
According to Deutsch, when photons interfere in the double slit experiment there are tangible photons and shadow photons involved. The tangible photons are the ones that the experimenter sees and the shadow photons are in the parallel worlds. They only interact to make the interference phenomenon work. Otherwise, they cannot be detected.
What that means is that these worlds must be synced together so that the interference can occur or not occur depending on whether the experimenter opens both slits or covers one of them up. Deutsch claims this about parallel universes (page 51):
It is the explanation--the only one that is tenable--of a remarkable and counter-intuitive reality.
I think Penrose has a way to consider photons being in two places at once without requiring parallel universes, so this is not the only tenable explanation regardless of Deutsch's claim. If Penrose is right, his approach would also lead to a deeper understanding of the universe and not just compound the counter-intuitiveness of reality.
But in any case the many worlder's need an experiment to prove that these other universes exist. I think the following would work.
Shine a single photon through a double slit until it reaches a photomultiplier that will signal a computer to generate a random number. This random number will be the seed to a series of random numbers that will either cover the second slit or open it depending on whether the random number is even or odd. If the slit is covered there should be no interference. If the second slit is open, interference should be detectable. Set up a mechanism to automatically register whether the interference pattern occurred when it was expected to occur or not and look for a situation when the interference pattern was missed. Automate the test so it can be repeated as rapidly as possible to generate many tests results.
If there really are other universes the interference pattern should be missed every now and then. That would be the proof that the many worlds exist although it would contradict the double slit experiment's expected results.
If the interference pattern is never missed the many worlders then need to explain how this counter-intuitive synching process occurs. That would not prove they are wrong, but it would make one wonder what difference it makes to compound a simple counter-intuitive observation with an explanation that is way more counter-intuitive and which cannot be detected by experiment.