Demon Parts are items in Okamiden that Chibiterasu can use, along with Lucky Coins, to upgrade his Divine Instruments. There are three different types, Demon Skin, Demon Liver and Demon Bones and they can be obtained from using Floral Finishers on various Demons. Chibiterasu can see how many of each he has collected from his menu screen.
He can then use them to upgrade his Weapons at the Blacksmith in Yakushi Village. The Blacksmith will appear after he has convinced at least one person to move into the village. Demon Bones can be gotten at the start of the game however Demon Skins cannot be gotten till Ryoshima Coast and Livers cannot be gotten until Chibiterasu travels back in time. In total 45 Demon Bones, 45 Demon Skins and 40 Demon Livers are needed for all Weapons upgrades. Apart from the Purity Reflector, all Weapons require two type of Demon Parts, so they cannot be upgraded till Chibiterasu reaches Ryoshima Coast.
An Old Man from South Ryoshima Coast will Migrate to Yakushi Village once the hot spring has been built. He will eventually buy Demon Parts off Chibiterasu for Yen. He can be found on the second floor of Naguri's inn through the entrance to the right.