Sprinkle chicken pieces with 1/2 teaspoon salt and 1/4 teaspoon pepper. In a Dutch oven, heat oil over medium heat. When it's hot, add chicken pieces, and brown well, turning them occasionally, about 15 minutes (if necessary, cook chicken in two batches to avoid crowding the pan). Transfer to a plate.
Drain most of the fat from the pan. Add onions, half the bell peppers, 1 tablespoon parsley, bay leaves, thyme, and oregano. Cook until onions are translucent, about 5 minutes. Add wine, and scrape up browned bits from the bottom of the pan with a wooden spoon.
Stir in tomato sauce. Return chicken to the pan, along with any accumulated juices, and add parsnips, carrots, and remaining bell peppers. Add enough water to barely cover; bring to a boil. Reduce heat, partially cover the pan, and simmer until chicken is almost cooked through, about 15 minutes. Uncover pan, and continue simmering until chicken is done and the liquid thickens slightly, 10 to 15 minutes. Stir in remaining 2 tablespoons parsley, and serve.
Rice traditionally accompanies this stew; mashed potatoes would also be delicious.