Combine the chicken, onion, garlic, whole carrot, whole celery stalk, bay leaves, and parsley, if you're using it, in a stockpot. Add 8 cups water. Bring to a boil, then lower the heat so that the liquid bubbles gently but steadily.
Cover and cook, stirring gently every 15 minutes or so, until the chicken is fully cooked (30-40 minutes for parts, 50-60 minutes for a whole one). If necessary, add water to keep the chicken submerged. To check for doneness, use a slotted spoon or tongs to carefully lift out some chicken and pierce it with a thin-bladed knife. You should meet no resistance, and the meat should be white all the way to the bone and may start to fall off in places.
Carefully transfer the chicken from the pot to a shallow bowl with tongs. When it's cool enough to handle, pull off and discard the skin; pull the meat off the bones and cut it into bite-size pieces. Reserve 2 cups for the soup and refrigerate the rest for another use. If you have time, return the bones to the pot and let the stock simmer for another 15-30 minutes.
Put a strainer or colander over a large pot and carefully pour in the stock; press down on the vegetables to extract as much liquid as possible, then discard them. Skim the fat off the top. You should have about 7 cups; if not, add some water. Sprinkle the stock with a little salt and pepper. (If you just want to make chicken stock, stop here.)
Put the pot over medium-high heat. Bring it just to a boil, then lower the heat so the liquid bubbles steadily. Add the chopped carrots and celery, and cook, stirring occasionally, until the vegetables are as crisp or tender as you like (anywhere from 10-30 minutes).
Stir in the pasta and the reserved 2 cups chicken; adjust the heat so the mixture bubbles steadily and keep cooking until the pasta is tender but not mushy (another 5-10 minutes). Taste and adjust the seasoning, garnish with the reserved celery leaves, and serve.