Link's parents...
Link is an orphan in all games that address his family.
In OoT the Great Deku Tree explains to him that he is not a Kokiri, but a Hylian - that his father was killed in battle and his mother, while wounded, fled to the forest to save him as a last act.
In LoZ/AoL... the games do not address Link's family, but he has parents in the NON-CANON North American comics - they're winemakers in the neighboring country of Calatia. He visits them in a couple of the comics and they look to be very honorable people. Still, the games, the actual canon does not address this - in them, Link is just a wanderer who found Impa, learned the legend of the Triforce and of Zelda and so starts the 8-bit adventure.
In A Link to the Past - he has an uncle who (in the mangas is killed in the castle), but in the game, I believe he lives, he just gets wounded and you can find him under a bridge later in the game where he gives you a bottle and falls into a deep sickness-sleep. When you beat the game, your Triforce wishes bring him back along with healing the land.
In Twilight Princess... Link's parents are nonexistant. We just don't see him having *any* family, just a mentor and his family whom he's close with. You can say he has a family-of-sorts with Ordon Village, but he's the only pointy-eared Hylian there with no sign of blood relations. I wrote my own tragic fanfiction on the subject, but since we're speaking of canon - no we don't know what happened to them.
In Wind Waker, Link has a grandmother and a sister. No parents. I just assumed they'd died at some point, as I do with all the parents of the Links.
I think it's become a tradition with Nintendo to make Link an orphan. When you think about it, it makes him more free. Do you know what used to be on old west Pony Express posters when they sought young men to ride over dangerous country to deliver mail? "Orphans Preferred." (Because they knew many of these young men wouldn't come back and they didn't want to cause grieving family). The quest of every Link is dangerous (at least judging by how many times they've died when I've played them!). Parents would be fussy and wouldn't want to let him go upon the quest. The wisdom of a grandmother might say "He must go - he has to do what's right" while a mother would be inclined to be very protective "My baby's not going out there!" So, to me, it makes sense that Link has limited family and never any living parents by the time he must face his destiny.