In Star Trek: Voyager (Scorpion, Part II) Janeway and crew travel to Fluidic Space, the home of Species 8472.
Do they reside in a parallel universe or some other dimension altogether?
Based on the fact that you need to traverse an "interdimensional rift" in order to get into Fluidic Space, it would seem highly likely that it's an alternate dimension rather than a mere subspace domain.
TUVOK: We appear to have crossed an interdimensional rift...
SEVEN OF NINE: My subspace link to the Collective has been weakened by the interdimensional rift. We cannot signal for help. We are alone. (beat) We must construct a compliment of bio-molecular warheads and modify your weapons to launch them.