Who was the father of Thorin Oakenshield?

The father of Thorin Oakenshield was Thráin II.

Thráin II was a Dwarf of Durin’s Folk and the son of Thrór, the King under the Mountain. Thorin was therefore Thrór’s grandson. This family line matters because Thorin’s claim to Erebor came through his royal descent, even after Smaug drove the Dwarves from the Lonely Mountain.

After the loss of Erebor, Thráin and his son lived in exile. Thráin eventually became consumed by the desire to reclaim his kingdom and recover the last of the Seven Rings of the Dwarves. He disappeared while travelling toward Erebor and was later imprisoned in Dol Guldur, where Gandalf found him shortly before his death.

The names are easy to mix up. Thrór was Thorin’s grandfather, not his father; Glóin was a member of Thorin’s company and later the father of Gimli; and Dáin II Ironfoot was Thorin’s cousin. Thorin himself is also called Thorin II, which distinguishes him from the earlier Thorin I.