First, they appear to Guybrush after he knocks his head and falls unconscious at the base of the big tree on Booty Island. Guybrush asks them why they left him, but they claim that they were actually looking for him. They then offer him some advice, and rip their skin off, leaving them as skeletons. Guybrush is horrified, but he is intrigued by their song and dance, and writes down the bones they list in a particular order. They run away when the Zombie Pirate LeChuck approaches and ends the dream even more strangely than it started. Guybrush doesn't think much of the vision but he does have the instruction written down on the spit-encrusted paper, and it later proves very useful in navigating LeChuck's Fortress.
At the end of the game, Guybrush finds the skeletal remains of his parents in an underground tunnel, the cause of their deaths unknown. After LeChuck is dying and claims to be Guybrush's brother Chuckie, Guybrush finds himself, a young boy, at an amusement park with his parents. However, Elaine suspects that Guybrush seems to be under a spell, and when we next see him (in The Curse of Monkey Island), there is no sign of this world, so the reality of this scene is in doubt.
In Escape from Monkey Island, the waitress at Planet Threepwood names Guybrush's parents as Bertha and Dwayne, but none of the other details about Guybrush's personal life are accurate, so there's no proof that these are their real names.
Guybrush mentions that his parents had abandoned him sometime when he was young, though they did seem to give guidance in the form of advice, which is frequently mentioned in Tales of Monkey Island. His father is mentioned in "Lair of the Leviathan", when Guybrush, while being chased by the Democratically United Brotherhood of the Manatee Interior and asked by Morgan LeFlay what she wants, tries telling her she needs his lucky ankle warmers and that she "Can't run a man through with cold ankles, that's what my Pa always said"; and his mother is mentioned twice: once in "The Siege of Spinner Cay", when he taunts Captain McGillicutty with one insult that says, "My possibly dead mother could shoot a cannon better than you can!", and once towards the end of "The Trial and Execution of Guybrush Threepwood", when our hero has been mortally wounded in a blow dealt by LeChuck's Cursed Cutlass of Kaflu and knocked unconscious onto an encased wind idol in the Flotsam Jungle, and Elaine makes her desperate attempt to revive him, at which point he wakes up and sees her in a daze as he mumbles, "Is that you, mother? I washed my hands..."