Link's father was killed prior to the game's events, leaving Link to be raised by his Uncle. As the game progresses it is hinted he might still be alive. When Link finds himself in the Village of Outcasts, in the Dark World he is told that they had passed through there. When Link sees Agahnim's memories in Ganon's Tower he discovers that his father and Agahnim were once friends but in a desire for power Agahnim betrayed Link's Father and sent him to the Dark World. It is further revealed that his parents are now dead. After Link defeats Ganon and receives the Triforce he does not wish for his parents to live again although the thought does cross his mind. He instead sacrifices his own happiness for the sake of Hyrule by wishing to revert all that Ganon did to the world.
In the Ocarina of Time manga, Rauru, the Sage of Light, tells Link that he was indeed born into the house of guardians that protected the King of Hyrule and his family, and that his father was one of those guardians. He goes on to tell Link that his father was killed in battle during the Hyrulean Civil War. This explains his absence in the game, as well as the Hero of Time's blood link to the Knights of Hyrule, as stated by the Pedestal of Time in A Link to the Past.
In A Link to the Past, it is said that the Knights of Hyrule have died out, but implied that the Link in this game is their descendant. In fact, the Nintendo-licensed graphic novel for A Link to the Past implicitly states that Link's parents were Knights of Hyrule. Whether this means that the Links from Ocarina of Time and A Link to the Past are blood relatives is, while plausible, only speculation.
Link's Father is a man named Arn, who lives in Calatia with the rest of Link's family.
Although never confirmed, it is possible that Link's Father is first referenced in A Link to the Past, as the Pedestal of Time reads:
The Hero's triumph on Cataclysm's Eve,
wins three Symbols of Virtue.
The Master Sword he will then retrieve,
keeping the Knights' line true.
Link from The Legend of Zelda: Ocarina of Time is presumably the hero mentioned in the above narrative. As a child, he won the three Spiritual Stones: the Kokiri's Emerald, Goron's Ruby and Zora's Sapphire. On the day he completes this quest and returns to deliver the stones to Princess Zelda (the day referred to as the "Great Cataclysm's Eve"), he unwittingly lets Ganondorf into the Sacred Realm by opening the Door of Time, allowing Ganondorf to seize and inadvertently shatter the Triforce. This Link awoke seven years later and retrieved the Master Sword, using it and the power of the Seven Sages to fight Ganondorf/Ganon. The statement that he "kept the Knights' line true" implies he was descended from the Knights of Hyrule, and that possibly, his father was one of these knights.
In The Legend of Zelda: Breath of the Wild, Link's father is briefly mentioned by Zelda in a recovered memory; however, the only information she gives is that his father was indeed a Knight of Hyrule, and that Link seems to be following in his footsteps.