A few answers come to mind:
- There's the Mirror Universe from TOS, which reappears in later series. It seems to be connected to the primary universe in some way.
- There's the antimatter universe, from TOS.
- There's a near-unlimited number of quantum realities seen in TNG 7x11, "Parallels".
- Then there's the alternate timeline created for the 2009 movie.
As far as I know, it is unknown if there's any overlap from these. The Mirror Universe, Antimatter Universe, and 2009 Movie may or may not be a quantum reality from Parallels.
And in addition to the above, which specifically fall under "parallel universe", there are other dimensions/universes that seem unlikely to fall under that umbrella term: Fluidic Space, the Q Continuum, and the one Dax accidentally created, for example.