I don't think you can get bones without killing a dragon, but there is one place where you can get scales. On the hill north/northeast of Rorikstead, there is a shrine to Akatosh that has a few "offerings" lying around it. One of these is a dragon scale. If you leave the area and wait for 10 days, the items will all respawn, meaning it is possible to craft and upgrade a full dragonscale set before doing the Dragon Rising part of the main quest.

I don't think you can get bones without killing a dragon, but there is one place where you can get scales. On the hill north/northeast of Rorikstead, there is a shrine to Akatosh that has a few "offerings" lying around it. One of these is a dragon scale. If you leave the area and wait for 10 days, the items will all respawn, meaning it is possible to craft and upgrade a full dragonscale set before doing the Dragon Rising part of the main quest. If you have the Dawn Guard DLC you will find one dragon bone in the laboratory of Serana's mother.