At the moment, Marvel and DC are both doing big superhero crossovers about multiple parallel worlds colliding and fighting. There are a few settings out there with fleshed-out parallel worlds, so how would a game with a sufficiently high Weird Level cope with a bunch of them turning up at once?
For a more isolated example, Stargate SG-1 covered this in one episode, and it seems none of the visiting parallels had been established in the series’ previous parallel-related episodes.
If you have existing parallels bring them in. If you need more, sketch out some options. What would a Silver Age version of your modern supers game be like, for example?
As a GM, I’ve seen one very effective way to quickly make a parallel feel different... shuffle the PCs. Player 1 will player Player 2’s character differently even if the rest of the universe is the same. (Not to be tried if the players will resent the resulting characterisations of their PCs...)
A game all about warring parallels is possible as well. See GURPS Time Travel, and Fringe... In this case it could be interesting to let the players create their alternate selves and play them in some sessions, to ground the different world.