Wash the chicken pieces and place them in a saucepan with the water. Add the onion, celery leaves, parsley, bay leaf, allspice, salt and pepper. Simmer, covered, for two and one-half hours.
Lift chicken and bones from the stock and strain liquid through a fine sieve. The stock may be further reduced if a concentrated flavor is desired. Cool the stock and chill it. Discard the layer of fat before using.