Pull and cut skin from duck breasts and legs. Slit leg skin open to lie flat. Put skin in a single layer in a pan about 10 by 15 inches.
Bake in a 350° oven until skin is crisp and golden, about 1 hour, turning pieces over occasionally. Drain skin on towels. Scrape duck fat from pan into a bowl and reserve; wipe pan clean.
Meanwhile, mix salt and sugar and rub heavily all over breasts and legs. Put duck in a bowl, cover, and chill at least 30 minutes but no longer than 1 hour.
Discard liquid that forms and rinse duck well under cool running water, massaging to release salt and sugar. Put duck in the 10- by 15-inch pan. Sprinkle with thyme and pepper. Add 1/4 inch of water to pan. Cover pan tightly with foil.
Bake in a 325° oven until meat is very tender when pierced, about 1 1/4 hours.
Meanwhile, in a 10- to 12-inch frying pan over medium-high heat, combine bread crumbs and 2 tablespoons reserved fat from duck skin. Stir until bread is richly toasted. Pour into a bowl. Coarsely chop duck skin.
Transfer cooked duck to a large platter and keep warm. Put duck skin cracklings on a piece of foil and return to oven to heat, about 5 minutes. Pour into a small bowl.
Sprinkle toasted bread crumbs and parsley over duck. Add cracklings to portions as desired.
Per serving without cracklings: 227 cal., 38% (87 cal.) from fat; 25 g protein; 7 g fat (6 g sat.); 5 g carbo (6 g fiber); 407 mg sodium; 92 mg chol.
Roasted cracklings: Nutritional information not available.