Parallel universe

As Sergeant John Benton once observed, a parallel universe was "some sort of mirror universe, like in that Star Trek episode where Spock had a beard." Amused, called Benton's assessment "succinct." His own belief about parallel universes was more complex; to him, a parallel Earth occupied "the same space-time coordinates as this Earth, but in a different dimension. Sideways in time, if you like, rather than forward or back." (PROSE: The Face of the Enemy)

Physics
As the Fifth Doctor once explained to Turlough, parallel universes were a specialised form of alternate universe — "alternative universes [that existed] in parallel with each other" — but the terms weren't complete synonyms. A parallel universe had no impact upon another parallel universe; it was "separated only by a millisecond of time and a nanometre of space, without ever having contact." Conversely, a fully alternate universe was when, because of the inherent weakness at the point that one universe split from the other, one timeline actually destroyed the other. (PROSE: Imperial Moon)

At least some parallel universes were created as a consequence of people making different decisions — the Tenth Doctor explained to Jackie Tyler that "every single decision we make creates a parallel existence". (TV: Doomsday) The Tenth Doctor also said that there were "billions of parallel universes all stacked up against each other. The Void is the space in between, containing absolutely nothing." (TV: Army of Ghosts)

Time behaved differently in some parallel universes compared to N-Space. In one universe the Inferno Project was more advanced. (TV: Inferno) Pete's World was three years ahead, (TV: Doomsday) while William Blake spent a few days in Yssgaroth's universe and returned to Earth seventy years later than when he left. (PROSE: The Pit)

At one point, travel between parallels was made relatively simple by the Time Lords, but, according to the Tenth Doctor, travel to parallel universes became almost completely impossible after the Last Great Time War, and could only happen under extraordinary circumstances. (TV: Rise of the Cybermen)

Examples
In July 1963, the Fifth Doctor visited a parallel universe in which World War II was still being fought. While there, he met Gus Goodman, who became his companion. (COMIC: Lunar Lagoon, 4-Dimensional Vistas, The Moderator)

In the 1970s, the Third Doctor visited Inferno Earth in which the United Kingdom was a fascist Republic ruled by an alternate Doctor. (TV: Inferno, PROSE: Timewyrm: Revelation)

Circa 1979, the Fourth Doctor visited a universe in which the Roman Empire spanned the galaxy and was beginning to conquer other universes. (COMIC: The Iron Legion)

One parallel universe was lifeless and empty due to the destruction of Logopolis in 1981. (PROSE: The Quantum Archangel)

In the 1990s, Carbury was invaded by knights from a parallel universe which bore similarities to Arthurian legend. (TV: Battlefield)

The Sixth Doctor and Evelyn Smythe travelled to a parallel universe in which the Doctor actually enjoyed his exile on Earth with the Brigadier. (AUDIO: The 100 Days of the Doctor)

In 2007, the Tenth Doctor, Rose Tyler and Mickey Smith visited a universe in which Rose's father was still alive and the Cybermen were created on Earth by John Lumic. In this universe, the United Kingdom was instead the People's Republic of Great Britain led by the President of Great Britain. (TV: Rise of the Cybermen) The Torchwood Institute in this universe had managed to develop a technology that allowed quick travel between their dimension and N-Space through the breach created in the Void by the Dalek Void Ship, but as an unintended side-effect, travelling in this manner further opened the breach, to a point where it started threatening the existence of that universe. When the breach was closed by the Doctor, this method of travel was rendered unusable. (TV: Doomsday)

In 2008, Ianto Jones accepted the offer of Mairwyn. He did not die when the 456 invaded and Mairwyn enslaved Earth. The two later visited the primary universe to observe Ianto's funeral. (COMIC: Shrouded)

A parallel universe, where Earth was part of a meta-government called the United Federation of Planets, was joined with the Doctor's at one point, resulting in the Fourth Doctor coming in contact with the Federation in the 23rd century and the Eleventh Doctor in the 24th century. (COMIC: Assimilation²)

"All thirteen" incarnations of the Doctor came together to save Gallifrey in the Last Great Time War by sending it into a parallel pocket universe. (TV: The Day of the Doctor)

Miscellany
Fitz Kreiner once told Trix MacMillan that "the future" was the best place to shop for music, since one could pick up an artist's complete run. Parallel universes, however, complicated this theory: "You think you've got all the Beatle albums until you've been to a few of those." When later called upon to sing in a bar, Fitz briefly considered using one of the "parallel universe Beatles songs," since no one in the prime universe would recognise them. However, when Trix pointed out that one of these songs was merely "Jealous Guy" with different lyrics, he quickly abandoned the idea of using these songs as "cheating." (PROSE: The Gallifrey Chronicles)

Chronovores feasted on parallel universes, but their meals were infrequent because they were banished to Calabi-Yau Space, which provided them no access to the timelines and parallels of the normal universe. once released the Chronovores to N-Space, where they threatened to eat away all realities. (PROSE: The Quantum Archangel)

As colloquialism
The term was used sometimes in an inexact way, simply to indicate the unlikelihood of an event. Claudia Bruderbakker once told Peri, for instance, that if her stepmother had been responsible for a particular event, she'd "entered a parallel universe". Peri, familiar with parallel universes, briefly considered the genuine possibility of another universe being created, but then realised that Claudia was speaking figuratively. (PROSE: Synthespians™)