One thought is that multiple dimensions does not at all necessarily mean parallel ones. The other universes sharing physical space with ours could be anything.
In String Theory, the hypotheses that there are actually 11 dimensions and parallel universes are really separate issues - and the 11 dimensions would actually exist in each parallel universe just as the 4 dimensions we're aware of (height, width, depth and time) exist here in this universe - separate universes wouldn't exist in the 11 dimensions. But to really understand all that, one would need to read up on String Theory.
Not to mention it's a discussion that goes way beyond the scope of this forum.
One of the more interesting theories for parallel universes is the one that I mentioned in the topic that I linked to in reply #11. It's fascinating to think that every time we're presented with a choice, the possibilty is that we actually make each choice and create an alternate/concurrent band of time where each choice is indeed lived out. And because there are an infinite amount of choices to be made, there are infinite parallel universes - though the bands in which the choices made correspond most closely would supposedly exist closest to each other, whereas the ones furthest away would be as different from our own universe as they could possibly be.
It's certainly fanciful but still pretty cool to fantasize that in some parallel universes DS was never canceled in 1971, it's still running on ABC, and our alternate selves might be coming to a forum similar to this one to discuss the current DS storylines rather than those from 40 years ago.