Final answer:
Thorin learns about Bilbo giving the Arkenstone to Bard and the Elvenking when Bilbo confesses during a parley, hoping to use the stone as leverage for peace negotiations.
Step-by-step explanation:
Thorin Oakenshield discovers that Bilbo Baggins has given the Arkenstone to Bard and the Elvenking during a parley when Bilbo admits to giving them the stone to use as a bargaining chip. Thorin is infuriated by Bilbo's decision to use the precious heirloom of his people in this way, feeling betrayed and enraged. Bilbo hopes that by surrendering the Arkenstone, which Thorin values above all else, it would help forge peace and avoid unnecessary bloodshed over the treasure hoarded in the Lonely Mountain.