"You must be good. My father only accepts the best and the brightest."
His most significant contribution was in command of the Pathfinder Project and its follow-up, Operation Watson, which developed major advances in long-range communication to successfully reestablish contact with the lost Starfleet vessel USS Voyager. (VOY: "Pathfinder")
Owen was an instructor at Starfleet Academy, having taught his son's course on Survival Strategies in the 2360s. Owen did not play favorites with his son as his student, giving Tom an overall "B-minus," in the course. (VOY: "Parturition") Owen often pressured Tom to excel in everything he did, frequently making his son feel inadequate. He continually talked about the Prime Directive to Paris and other Starfleet officers. To him, the Prime Directive was one of the leading principles in space exploration. (VOY: "Time and Again")
As captain of the USS Al-Batani, Kathryn Janeway served under him on the Arias Expedition. Later, when Janeway was assigned as the captain of the starship Voyager, she read a file on Paris' son, Thomas, and wanted him to come on Voyager's first mission as observer to help navigate the Badlands. She later brought up her past service with Owen Paris, when she went to meet Tom at the penal colony he was serving time at in New Zealand. (VOY: "Caretaker") Before venturing to meet with Tom Paris, she ran the idea past Admiral Patterson, who recognized the name as "Admiral Paris' son." (VOY: "Relativity")
According to Tom Paris, he never changed an opinion once the opinion was finalized. (VOY: "Hunters") Also according to Tom, "my father, he'd say crying is a sign of weakness," however, Tom ultimately "never believed that." (VOY: "Threshold")
In 2376, Owen Paris was overseeing the Pathfinder Project at Starfleet's Communications Research Center, which was aimed at establishing communications with the starship Voyager stranded in the Delta Quadrant. He had an office where he did his work, with his secretary Nicole. As the admiral overseeing the Pathfinder Project, Commander Pete Harkins reported directly to him, and he visited for inspections on progress routinely.
During such one inspection in 2376, Admiral Paris was being briefed on new plans underway, using the MIDAS array, by Commander Harkins when Lieutenant Reginald Barclay interrupted the presentation. He made a spectacle about a micro-wormhole they may be able to create, to achieve real-time communication with Voyager. Harkins shot down the idea, but Barclay later went around him directly to the admiral at his office, "scaring [his] secretary." The admiral agreed to look over Barclay's findings, and sent him away.
When Paris tried to contact Barclay to say his findings may have merit, the admiral was disappointed to find that Barclay had tried to go along with them without authorization. Before he could punish him, however, the communications signal went through and Voyager began to respond. Admiral Paris was able to reassure Captain Janeway that they were still working to bring the Voyager crew back home, and had Barclay send tactical and communication updates before the link ended. (VOY: "Pathfinder")
Several months later, in 2377, after two months of data-streams to Voyager failing to send a complex hologram, Admiral Paris was called in regards to a breach of Pathfinder security. Along with Counselor Deanna Troi and Commander Harkins, they questioned a woman named Leosa about the loss of the Barclay hologram. Leosa claimed to be a teacher while dating Barclay but was, in fact, working for the Ferengi. Though Leosa claimed that she knew nothing about this, Admiral Paris theorized that the Ferengi had somehow found a way to profit from the hologram and that she secretly knew this. His theory later proved correct, when she revealed the ship, though not its plan. Admiral Paris then ordered Captain Peterson, of the USS Carolina, to a red giant star in sector 39542 grid 8, proceeding at maximum warp to apprehend the Ferengi. (VOY: "Inside Man")
The following year, Admiral Paris was at Pathfinder labs for the testing of Operation Watson, which allowed for eleven minutes of daily contact with Voyager. Days later, Lieutenant Barclay confronted him about a holoprogram he'd become aware of that depicted the Voyager crew in an unflattering light. The admiral immediately contacted Captain Janeway about it, and a legal battle soon ensued. He sat in at a subsequent Federation judicial arbitration between Voyager's EMH, "The Doctor," and Ardon Broht of Broht & Forrester Publishers, over The Doctor's rights to own property. (VOY: "Author, Author")
While Paris and several other admirals were at the Project Pathfinder labs, Barclay reported that their sensors detected a Borg transwarp conduit opening less than a light year from Earth. Fearing another Borg invasion attempt, he ordered all available ships to converge on the coordinates of the conduit. Commanding the fleet of eighteen ships, with nine more on the way, he ordered for them to use all necessary force to stop any Borg ships. A Borg sphere eventually emerged, but began to explode on its own and Paris ordered the fleet to hold their fire as the sphere exploded, with Voyager emerging triumphantly from the wreckage. Admiral Paris hailed Voyager immediately, to welcome them home. (VOY: "Endgame")
Admiral Paris appeared wearing the rank insignia of a three-star or vice admiral in "Persistence of Vision", then a four-star admiral in "Pathfinder", then three-star again in "Inside Man." This could mean he was a vice admiral before Voyager left, and was promoted while they were in the Delta Quadrant to full admiral. The change in "Inside Man" was likely a costuming mistake, or he could have been demoted. But later in both "Author, Author" and "Endgame", he wears the four star insignia again.
In a deleted scene cut from the final draft of VOY: "Friendship One", Tom Paris complained that "the only people who want to talk to me [from the Alpha Quadrant] are my father and a Ferengi Tongo dealer who claims I still owe him three hundred strips of latinum."
In Star Trek Beyond, a character named Commodore Paris is in command of Starbase Yorktown, with Simon Pegg confirming that her name is a reference to Tom Paris, suggesting that she is the alternate reality counterpart to Tom's grandmother, and hence Owen's mother. 
Star Trek: Voyager co-creator Jeri Taylor's two novels, Mosaic and Pathways, explore Captain Janeway's early career, especially her time on the Al-Batani with Captain Owen Paris, during the Arias Expedition. In the novels, the two met in 2355 when she was still in Starfleet Academy, eventually handpicking her for his ship and the mission.
Several novels expand on his family and ancestry as well. In Serpents Among the Ruins, his father, Michael Thomas Paris, (an Academy classmate of Demora Sulu and "official" casualty of the Tomed Incident, though in truth he had been killed during a Special Ops mission in 2310 and, like all of the thousands of supposed victims from numerous bases and USS Agamemnon destroyed in the catastrophe, Starfleet covered up his death until the incident and falsified records to claim they had died there, in his case he had been aboard the Agamemnon) was a character and Owen's mother, Victoria Santos, was referenced. The Star Trek: Stargazer novel series features Owen Paris' brother, Cole, who was a crewmate of Jean-Luc Picard during his command of that vessel and entered into a complex relationship with a non-corporeal member of the crew after he accidentally walked into her while in her natural state. The Strange New Worlds IV story "Uninvited Admirals" names Owen's wife (and Tom's mother) Cinda.
In the Voyager Relaunch novel Homecoming, Owen and his wife (there named Julia) allowed Paris, Torres and their daughter Miral to live with them after they returned from the Delta Quadrant - as suggested in "Endgame".