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, the Master 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) However, the Eighth Doctor noted that such significant events remained indeterminate until the act of observing them forced the universe to "choose" the outcome, with the new timeline only splitting if absolutely necessary and transit between the universes normally being very difficult. (PROSE: Time Zero)
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 the Spiral Yssgaroth and returned to Earth seventy years later than when he left. (PROSE: The Pit, The Book of the War)
At one point, travel between parallels was made relatively simple by the Time Lords- although TARDISes were still equipped with security protocols to stop them travelling into other universes on a regular basis (TV: Inferno, PROSE: The Quantum Archangel)- 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 Angus Goodman, who became his companion. (COMIC: Lunar Lagoon, 4-Dimensional Vistas, The Moderator)
In the 1970s, the Third Doctor visited a parallel universe in which the United Kingdom was a fascist Republic ruled by an alternate Doctor. (TV: Inferno, PROSE: Timewyrm: Revelation)
The Fourth Doctor visited a universe in which the Roman Empire spanned the galaxy and was beginning to conquer other universes. (COMIC: Doctor Who and the Iron Legion)
One parallel universe was lifeless and empty due to the destruction of Logopolis in 1981. (PROSE: The Quantum Archangel)
In 1997, Carbury was invaded by knights from a parallel universe which bore similarities to Arthurian legend. (TV: Battlefield) The Eighth Doctor eventually returned to this world - before this invasion from their perspective - when he unintentionally disabled some of the TARDIS's security protocols, allowing him to set up the events that would lead to his past self thwarting their earlier invasion. (PROSE: One Fateful Knight)
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, although this combination was undone when the Doctor and the Enterprise-D defeated the Borg-Cybermen alliance that caused this invasion. (COMIC: Assimilation²)
In one universe, the Doctor stayed on Gallifrey and became an author. (AUDIO: Auld Mortality, A Storm of Angels, PROSE: The Innocents)
In another, the Doctor regenerated into a female incarnation to escape the Time Lords. (AUDIO: Exile)
Bernice Summerfield was once accidentally 'abducted' by an alternate version of the Doctor from a world where he was exiled to Earth in 1997 rather than the 1970s; after this universe had been subject to a major war, the Doctor had hoped to use Benny's accumulated artron energy to escape his dying universe, but instead he dragged her into his world. (AUDIO: The Library in the Body)
In one universe, the Valeyard won the trial and successfully obtained the Sixth Doctor's remaining regenerations. (AUDIO: He Jests at Scars...)
In another universe, Julia Caesar became empress of the Roman Empire and had her life prolonged by an alien technology. By the 1900s, the Roman Empire owned sixty percent of Earth and Caesar was still alive. America and Australia were populated by "savages". During an unknown incident, this universe's version of the Sixth Doctor lost one of his eyes. Melanie Bush was a slave. (PROSE: Spiral Scratch)
Melanie Bush met a version of herself from a universe where Silurians never went into hibernation. In this universe, she was Melanie Baal, a Human-Silurian hybrid. (PROSE: Spiral Scratch)
In one universe, the Doctor believed that the ends justified the means. (AUDIO: Full Fathom Five)
When Sabbath's actions caused the barriers between various parallel realities to break down (PROSE: Time Zero), the Doctor visited several different worlds trying to restore the prime reality, including a world where a group had been formed specifically to kill Sabbath (PROSE: The Infinity Race), a world where the computer was never invented as part of the efforts of that world's Sabbath to 'protect' his Earth (PROSE: The Domino Effect), a world where the mysterious Eternines sought to drain Earth's energy to save themselves (PROSE: Reckless Engineering), and a world where time-travel tours were founded while Earth faced invasion by an alternate race of Martians (PROSE: The Last Resort).
"All thirteen" incarnations of the Doctor came together to save Gallifrey in the Last Great Time War by freezing it in a single moment in time, in a "parallel pocket universe". (TV: The Day of the Doctor)
Donna's World was a "great big parallel world" created around Donna Noble by a Time Beetle, where the Tenth Doctor died facing the Racnoss under the River Thames in 2007, and was subsequently unable to save the people inside Royal Hope Hospital who were taken to the Moon in 2008, or the Earth from being attacked by a replica of the Titanic, the Adipose or ATMOS in 2008-09. When the Donna of this world travelled back in time to make her younger self turn left at a junction on Little Sutton Street, both the Doctor and Donna believed that this world ceased to exist. (TV: Turn Left)
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. The Tremas Master once released the Chronovores to N-Space, where they threatened to eat away all realities. (PROSE: The Quantum Archangel)
In all alternate universes with Melanie Bush she grew up with her sister. In the prime universe Melanie accidentally killed Anabel when they were children. (PROSE: Spiral Scratch)
As colloquialism
The term was used sometimes in an inexact way, simply to indicate the unlikelihood of an event. Claudia Bruderbakker once told Peri Brown, for instance, that if her stepmother Joan Bruderbakker 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™)