In the largest pot you have, combine the chicken pieces, chicken wings, onions, celery, garlic, parsley stems, peppercorns, and 1 Tbsp salt. Add half of the carrots to the pot and cover with the water. Bring to a boil, then lower the heat to a simmer and cook, skimming off and discarding any foam that rises to the top, until the chicken breasts are firm to the touch, about 25 minutes.
Use tongs to remove the chicken breasts from the pot and set them aside in a bowl. Continue simmering the stock, stirring it every so often and skimming any foam that rises to the top, until everything in the pot has given up all of its structural integrity (the vegetables should be totally soft and the chicken should look well past its prime—this is all great, it means these things have given all of their flavor to the water) and the stock is a rich golden color, about 3 hours.
While the stock is simmering, let the chicken breasts cool to room temperature, and then discard the skin, remove the meat from the bones (discard the bones), and shred the meat. Set the meat aside.
Ladle the stock through a fine-mesh sieve into a clean pot (or, if you don’t have another large pot, ladle it into a bowl, clean the pot you started with, and return the stock to the pot). Discard the contents of the sieve (everything in it will have given all it can by this point).
Bring the stock back to a boil and season to taste with salt (be bold, it will need quite a bit!). Add the remaining carrots and the parsnips, lower the heat, and simmer until the vegetables are tender, about 10 minutes.
Add the reserved chicken breast meat to the soup and let it warm up for a minute or two. Ladle the soup into bowls, and top each with some of the chopped parsley and dill. Serve immediately.
Note: This soup is even better the next day. Do not discard the hardened fat that will have formed on top after the soup has been refrigerated. The rich pools of chicken fat on top of your soup are essential (at least in my book, but no hard feelings if you would rather discard the fat).