In an 8-quart stock pot or soup kettle, cover the chicken with water. Add the celery, carrots, onion and parsley. Bring to a boil. Reduce the heat and simmer for 1 hour and 15 minutes, covered.
Using a spider skimmer, remove the chicken and vegetables and place in a colander in the sink to drain. Use a fine mesh sieve to strain the stock into a large bowl. Return the strained broth to the soup kettle. Reduce heat to simmer and continue simmering the broth.
Pull the chicken off the bones and return to the soup kettle.
Add the carrots, celery salt, celery seed and pepper. Simmer for 30 minutes. You can do these steps the day before.