The King and Queen want to make an alliance with Yugopotamia through the marriage of Princess Mandie and Yugopotamian Prince Mark, so that both planets could form an alliance and destroy Earth. They were shown during a flashback sequence while Mark Chang explained his arranged marriage with Mandie to Timmy, Cosmo, and Wanda.
Mandie's mother looks exactly like her, although her attire is slightly darker, with greenish-yellow skin, lavender-colored hair, two bug-like antennas on her forehead and triangular teeth.
Mandie's father resembles a wolf-like creature of some sort. He has red eyes, and is covered in brown hair. He wears tall black boots and red clothing around his shoulders. He also has yellow arm bands and a blue and yellow speedo.
The King and Queen of the Boudacians are only seen in an explanation sequence by Mark Chang, who is telling of how his parents and Mandie's want to form an alliance to destroy the Earth with their combined forces. They do not have any speaking roles in the show, and are only seen in this very brief shot.
It's unknown what became of them following their daughter's arrest. Nor if Mandie's plan to take over Yugopotamia was their plan all along.