Twig's parents are two adult Deer-foxes, and the parents of Twig.
Twig's father is the alpha of the Deer-Fox Herd that Twig originally belonged to. He is notably larger than an average deerfox, and has bigger antlers than the other males in the herd. Twig's mother looks like the other female deerfoxes.
They are first seen in "Chapter 9: The Deerfox". A few years ago, they lost Twig when he ran off to rescue Hilda from an eagle while the rest of the herd returned to their home dimension. In the present, Twig reunites with them and, after saying goodbye to Hilda, returns to their home dimension with them. The two realize however Twig's heart is with Hilda, and allow him to go back to her.
In "Chapter 8: The Fairy Isle", they come to Twig's aid to help him rescue Hilda and Johanna from Fairy Country. They use their hard light bridges to lead their entire herd to the Fairy Isle, and fight off The Eyes. Twig's parents also free Phinium and Lydia, who had been ensnared by vines summoned by The Fairy Entity. After returning Hilda and Johanna home, they say goodbye to Twig and return to their own world.