A parallel reality is a different reality running concurrently with the prime reality.
In 2373, Tuvok experienced a breakdown on the bridge of the USS Voyager over a traumatic memory suppressed into the subconscious of an event Tuvok had not experienced. The Doctor surmised it could have been caused by any number of things, such as aliens with telepathic ablilities or a brief experience with a parallel reality. After describing these to Tuvok, The Doctor commented "the universe is such a strange place."
After initiating a mind meld with Captain Kathryn Janeway, who was acting as Tuvok's pyllora, they both discovered that his breakdown was in fact not caused by contact with a parallel reality, but rather a memory virus unknowingly transmitted to him by his former crewmate from the USS Excelsior, Lieutenant Commander Dmitri Valtane, in 2293 during a battle with the Klingons. (VOY: "Flashback")