When Lucca was a child, Lara wanted to dust one of Taban's inventions despite him telling her to stay away from it. However, her skirt gets stuck in the machine's conveyor belt and can't pull it out. The machine then activates and the conveyor belt starts pulling Lara into the machine. Lara tells Lucca to input the password to stop the machine, but Lucca does not know it. As a result, Lara loses her legs due to Taban's invention.
In the many endings of Chrono Trigger, he makes a guest appearance. During the Moonlight Parade, he can be seen drinking "Soda" on the ledge of the fountain in Leene Square. If Lara's Legs are restored, she begs him to dance with her. If not, she sits on the ledge of the fountain while he stands. Additionally, in another scene, Taban strikes the Sun Stone with his mallet to produce fireworks, while King Guardia XXXIII hangs the Nadia's Bell.
Taban and Lara are probably dead by the year 1020 A.D.:
Lucca's room is not a bedroom anymore.
The bed in "Lara's" room is a single bed. It was the case in CT too, but that was because of graphical limitations; they could have made a two-person bed in CC if it was still Lara's and Taban's room. The children refer to that room as "Granny Lara's old bedroom" and "Lucca's mother's old bedroom"
In the note for the L1-Triangle-R1-Triangle password, Lucca says "My father, Taban, would be proud."
Kid says "Lucca... All my friends...They're, they're...!!" She doesn't mention Lara or Taban.
Kid fends for herself after the fire; if Taban and Lara were alive, they might have taken care of her (though it is possible they assumed she was dead).