Rinse chicken and place in a large, deep saucepan. Add cold water and slowly bring to a boil over medium heat. Add onion, 2 stalks of celery, 2 carrots and peppercorns. Reduce heat and simmer, using a spoon or ladle to skim foam that rises to the surface. Stir in 1 tsp (5 mL) salt. Simmer stock, uncovered, for 2 hours, occasionally skimming fat and foam that collect on top.
Use a slotted spoon to transfer chicken and vegetables to a cheesecloth-lined strainer set over a large bowl. Carefully pour stock through strainer, allowing to drain but not pressing down on the solids as this can make the broth cloudy. Reserve chicken for another use; discard cheesecloth and vegetables.
Return stock to saucepan. Skim any remaining fat. Slice or dice remaining celery and carrots and add to broth, along with parsnips. Bring back to a simmer and cook about 20 minutes or until vegetables are just tender. Add fresh dill and additional salt, to taste.