It's definitely worth doing - they're basically really good Fuse materials for your weapons, and don't count towards the 'one per encounter' materials you get above! There's also Shards of the Dragon's Spikes on every dragon's back, little resources you can pick up in large numbers.
We're not sure what the cooldown is before farming another piece, but it's at least twenty-four hours in-game. You can only knock off one piece of a dragon per encounter - the first arrow that lands will be the last one of any use. Depending on where you hit it changes what kind of reward you get (though you'll need to dive down to the ground to grab it where it lands). When you encounter a dragon, hitting it with an arrow knocks off a piece of it to become a reward.
