Make the gravy
In a large (8- to 10-quart), heavy-bottomed saucepan or soup pot, melt the butter over medium heat. Add the flour and mix well to make a roux. Cook, stirring frequently, until the mixture is pale yellow and resembles fine, wet sand, about 3 minutes.
Whisk the soup into the roux a little at a time, allowing the roux to absorb the liquid before adding more (this will help avoid lumps). Add the salt, pepper, carrots, and celery. Bring the mixture to a boil over high heat, stirring occasionally, then lower the heat and gently simmer for 15 to 20 minutes. Keep stirring occasionally and skim off and scum that rises to the surface.
Make the dumplings
Whisk the flour, baking powder, salt, and parsley together in a large bowl, then cut in the butter using a pastry blender, two knives, or a whisk until it's in small pieces. (Alternatively, you can use a food processor: Place the flour, baking powder, and salt in a bowl of a food processor. Pulse several times to combine. Add the parsley and pulse once or twice to incorporate. Add the butter and pulse until the butter is in small pieces.)
Add the milk and stir or pulse to moisten the flour mixture. (Do not overmix or you will develop the gluten in the flour and the dumplings will be chewy.) Gather the dough into a ball.
Using a large spoon or your hands, scoop out 1/4-cup chunks of dough, lightly roll them between your palms to round them out, then drop into the simmering gravy (it's OK if they sink down), spacing them apart. Cover the pot and simmer until the dumplings are done (a knife inserted in the center should come out clean), about 20 minutes. (Avoid lifting the lid while the dumplings are cooking because it slows down the cooking process, and "if you're lookin' you're not cookin'!")
Gently stir the cooked chicken into the pot with the dumplings, return the liquid to a simmer, cover, and cook for 5 more minutes to heat the chicken through. Using a serving spoon or tongs, divide the chicken and dumplings among soup bowls. Ladle the gravy over the dumplings and chicken, sprinkle with the parsley, and serve.