Parker Killian, previously referred to as The Other Travis, Dark Travis, Mirrorverse Travis and Nega-Travis, is a version of Travis from a parallel universe where Travis was not plucked from time by Epoch Talent and consequently encountered the Wire, becoming her partner in business.
Alliance with the Wire
In an alternate timeline where Travis was not kidnapped and taken to the 51st century, Travis encountered the Wire in late 1998, finding her on a computer in the Sheffield Public Library where he worked. Together, they formed a company, Nest Online, to provide free internet to the people of Sheffield, NJ.
Eventually, Nerst Online grew large enough to be listed on the New York stock exchange, providing free wifi globally using a network of satellites, and introducing smartphone technology almost a full decade early. It is implied that Nest is also involved in the creation of the dating app, Matchmaker.
Using this technology, the Wire begins to take over the United States, turning the east coast into "a sea of wires, cables, junction boxes and machinery." (Both Ends Burning)
However, the Wire's goal was not conquest but to return back to her homeworld in search of revenge. She and the Other Travis build a satellite to broadcast her back out into space, to her home planet. Their plan is briefly interrupted, however, when the Travis from another reality replaced the Other Travis. The Wire was broadcast out into space, but died in transit due to the universe shrinking. Travis then used the dish to communicate with the Doctor and Millie in their universes, and returned back to his home universe. (Getaway)
But being placed in this universe converted Travis' mind to a more digital form, and the Other Travis' consciousness was trapped inside, piggybacking across into his other self's universe. He remained trapped there until Travis was taken to Gallifrey by Rassilon. He used the TARDIS' telepathic field to connect to the Matrix. Rassilon later discovered the Other Travis, and appointed him Keeper of the Matrix. (Both Ends Burning)
As Keeper of the Matrix
The Other Travis used this opportunity to build the Perfector, (Perfection) a device that would allow someone to sit inside of the Matrix and convert the universe outside into pure energy that could be consumed. While Rassilon intended to use this to become the only living being in the universe, the Other Travis intended to use the Perfector himself.
Rassilon, convinced of his inevitable victory, hung out in his throne room eating oranges. He was eventually stopped by the Doctor. The Other Travis, meanwhile, engaged the Perfector and prepared to use it himself, but was stopped by the original Travis, who erased the Wire's influence. Together, both Travises stopped the Perfector. (The Enemy of the Good)
Later, as the Time Lords instated the Doctorate Program, the Other Travis - now going by the name Parker Killian - was assigned to his own TARDIS. He entered for the first time alongside his teacher, Romana. Asking if he just went out looking for trouble, Romana simply responded, "Trouble's going to find you." (The Enemy of the Good)
Parker later appeared to assist Travis in preventing a nuclear meltdown at the Sheffield Nuclear Power Plant in 2002. He briefly succumbed to radiation poisoning, but was carried into his TARDIS and given anti-radiation drugs by Travis. Later, he called the Doctor's TARDIS to inform them that the Fatalists, who had planned the meltdown, still had an asteroid headed for Tunguska, Siberia in 1908. He Also revealed to Travis that their mother knew of his existence, though this is implied to have occurred in the relative future. (The End of History)