Cut 2 carrots into large chunks; cut remaining 4 carrots into thin slices and set aside. In a large soup pot, place carrot chunks, turkey carcass, onion, celery, parsley sprigs, bay leaf, thyme, and water. Bring to a boil over high heat; reduce heat to medium-low and simmer 1 hour, partially covered.
Remove turkey carcass from soup; pick off turkey scraps when cool enough to handle. Strain broth into a large bowl. In same soup pot, melt butter. Add carrot and leek slices; cook 5 minutes. Stir in beans, broth, turkey scraps, and parsley. Season with salt and pepper; cook on low 5 minutes.