In a medium bowl, combine the all purpose and whole wheat flours and the salt. Add the oil and stir well. Add the cold water 1 tablespoon at a time and stir until a dough just barely comes together. Place the dough on a piece of plastic wrap and wrap well. Refrigerate the dough while you prepare the stew.
Heat a stock pot over medium high heat. Add 1 Tablespoon oil and swirl to coat the bottom of the pot. Add the onion, carrot, and celery to the oil and stir until just softened. Add the chicken and stir until the color changes from pink to white, but not cooked through. Add the thyme and the hot stock and stir. Bring the stew to a simmer for ten minutes.
Remove the dough from the refrigerator and place onto a lightly floured counter. Dust the top of the dough with a little flour. Also flour the rolling pin. Roll the dough out to ¼ inch thickness. Cut the dough with a knife or rolling pastry cutter into ½ inch strips.
Drop the dough strips into the simmering stew and stir gently. Simmer, stirring every so often, for another 10 minutes or until the broth is thickened and the dumplings are puffed and the chicken is tender. Add salt and parsley and stir. Serve hot.