This page explains how to find the Farosh, Naydra, and Dinraal and farm scales, claws, fang shards, and horn shards from them.
How to Find Naydra
Before you can get dragon parts from Naydra, you have to rescue it first. Read the Jitan Sa'mi Shrine page to find out how to do that.
Afterward, make sure that you have some wood and some flint. You might also want to purchase the Snowquill armor set from Rito Village and have a Great Fairy upgrade each piece at least twice to give you the Unfreezeable set bonus.
Then go to Jitan Sa'mi Shrine, climb to a high point, and float north until you reach the base of the mountain where it borders Lanayru Bay. Start a campfire and sit there until morning. Then look west to see Naydra.
When Naydra comes near, float on the updraft and get as close as you can. Then if you want a Shard of Naydra's Horn, shoot Naydra's horn. If you want a Shard of Naydra's Fang, shoot Naydra's mouth. If you want Naydra's Claw, shoot Naydra's hand. If you want Naydra's Scale, shoot any other part of Naydra's body.
How to Find Farosh
Make sure that you have some wood and some flint. You might want to get the Rubber Armor set (from Toh Yahsa Shrine, Qukah Nata Shrine, and by talking to Cima at Lakeside Stable near Shai Utoh Shrine and completing her quest) and have a Great Fairy upgrade each piece at least twice to give you the Unshockable set bonus.
Then go to the top of Lake Tower and float from there to the eastern tower of the archway at the south end of the Bridge of Hylia. Start a campfire and sit there until night. Eventually, usually around midnight, Farosh will come out of the west side of the lake.
When Farosh comes near, float on the updraft and get as close as you can. Then if you want a Shard of Farosh's Horn, shoot Farosh's horn. If you want a Shard of Farosh's Fang, shoot Farosh's mouth. If you want Farosh's Claw, shoot Farosh's hand. If you want Farosh's Scale, shoot any other part of Farosh's body.
How to Find Dinraal
Make sure that you have some wood and some flint. You might want to get the Flamebreaker Armor set in Goron City and have a Great Fairy upgrade each piece at least twice to give you the Fireproof set bonus.
Then go to the top of Hebra Tower and float northeast to the cabin down below. Go inside and you can sleep in the bed (or just sit by the fire) until morning. Then go outside and go east to the cliff edge. Then look northeast to see Dinraal.
When Dinraal comes near, float on the updraft and get as close as you can. Then if you want a Shard of Dinraal's Horn, shoot Dinraal's horn. If you want a Shard of Dinraal's Fang, shoot Dinraal's mouth. If you want Dinraal's Claw, shoot Dinraal's hand. If you want Dinraal's Scale, shoot any other part of Dinraal's body.