One of the regular properties of particles is that they have a fixed position in space, while waves can occupy more than one place as they are vibrations not things. When relativity combines these two ideas we get the Heisenberg's Uncertainty Principle. This principle, the basis of Quantum Mechanics, states that we can never know the exact position and speed of a particle at any given time. Instead we can give all possible answers to both and suggest that the particle is neither doing one or the other (it has a "quantum state"). Richard Feynman took this one step further with his sum over histories theory that said that a particle gets from one place to another in all possible paths. Thus, with infinite possibilities for the location and movement of particles at any given time an infinite number of choices is available for the universe at any given time. And so with each choice a whole new universe for both outcomes exists in parallel.
This may sound all very academic to you but, unlike wormholes which are much more theoretical, there are indeed experiments to show not only the wave and particle nature of light but its ability to exist in two points in space time and yet exist in neither. It has a quantum state that incorporates all of these possible positions as a description of the state of the photon at any given moment. In the famous "two slit" experiment, particles are sent through two slits and hit a screen a distance away. When there are two slits the wave nature of particles causes interference and a pattern of dark and light spots is observed. However, since only one slit causes even distribution on the screen, adding another should not change this. Since it does, the particles are traveling through both slits at once. In this experiments the ability of the universe to have an infinite set of states at any given moment shows the existence of parallel universes.
As of yet, parallel universes of "Star Trek" have not yet been visited. While it is possible for entire other universes to exists with even reverse arrows of time, entropy, and pyschology, the problem remains in getting there. Every point of time in our universe produces an infinite number of possibilities and thus and infinite number of alternate and parallel universes branch off from every point. As I suggested in the section on wormholes one theoretical way of traveling to a parallel and opposite universe is to fall into a wormhole with one end rooted in the space-time of one universe and the other end in the fabric of another. This type of wormhole was hypothosized by Einstein and Nathan Rosen and the bridge point between two parallel universes in named an Einstein-Rosen bridge. It has been the subject of many science fiction plots as it theoretically allows both time travel to the past, future, and alternate time lines, a very handy tool for story telling.
Since we are unable to find any wormholes as of yet, and even if we could we would still be presently unable to overcome the space-time problems, the existence alternate dimensions and universes will remain a purely theoretical problem. Perhaps someday this knowledge will be of use to us, but for now the only things that have the privilege of being in more than one place at one time are the particles that make up our universe.