Soup: place the chicken, onions, carrots, celery, parsnip, parsley and salt and pepper to taste in a Dutch oven or large saucepan. Add enough water to cover the ingredients, about 10 cups, and bring to a boil over high heat. After 15 minutes, skim off the foam, reduce the heat and simmer 1 hour, skimming the soup occasionally as it simmers.
Dumplings: combine the flour and baking powder; beat the egg yolks on high speed until light, 3 minutes.
Add the parsley, mushrooms, flour mixture, salt and pepper; mix on medium speed for 1 minute.
In a separate bowl, beat the egg whites on high speed until they're stiff but not dry, 2 minutes.
Fold them into the yolk-mushroom mixture; set the batter aside.
Remove the chicken pieces and, when they're cool enough to handle, separate the meat from the bones and cut it into small pieces.
Return the chicken to the pot and bring the soup back to a simmer.
Drop the Mushroom Dumpling batter by the tablespoon into the soup.
Cover the pot and simmer the soup until the dumplings rise to the surface, 10 minutes more.
Serve the soup in warm bowls with a dumpling or two floating on top of each serving.