Quick travel to the Dow Na'eh Shrine (located almost directly in the center between Kakariko Village and Mount Lanayru behind a waterfall) and head north-east toward Lanayru Road – East Gate. Head to the arch / pillar closest to the east gate that goes into the beginning of the water on the east side (can be seen on your map). Climb the bluff directly north of it, and wait. The dragon will reach your point around a minute and thirty seconds after 9:10pm (or 12:10am). After you get the part you can either wait three minutes, or immediately drop wood to wait until 9:00pm again. There's two reasons for using the pillar closest to the East Gate: firstly, is that if it's raining, you can still light a campfire under it, and secondly, the dragon will be at face level (you on its' right side) from the bluff positioned above it, which causes the drop to fire directly beside you when you hit the dragon instead of way further west down the gorge.
I wanted to add, for min-maxing, if you've got extra wood you can sleep immediately after you collect the dragon drop to have the dragon of the next day spawn. Dragons spawn at 9:10pm, so if after ten seconds you don't see the dragon, one of two things has happened: either you already got the dragon drop for that day, or the dragon has a no-spawn glitch (which can be fixed by just saving and reloading that save, causing the dragon to spawn as soon as the game loads).
If you're low on wood, you can actually get two dragons within three – four minutes, by waiting until night, having a dragon spawn at 9:10pm, and getting its' drop before midnight. Once the clock hits 12:00am (00:00 military time) the dragon will respawn, meaning you can get two within four minutes (one at 9pm, three minutes getting the drop and waiting, followed by an extra minute or so for the new spawned dragon to reach your location again).
This will count as the dragon for the current day, however, so if you do this, you'll need to wait at the campfire twice instead of just once.