Place the chicken breasts in a soup pot and cover with cold water by 2 inches. Bring to a boil, skimming any froth, then reduce the heat to low. Add the onions, celery, carrots, salt, peppercorns and Parmesan rind, and simmer, partially covered, until the chicken is cooked through, 1 hour. Remove the chicken and set aside. Continue to simmer the soup, uncovered, for 1 hour longer, then remove the carrots and onions and set the vegetables aside.
Strain the broth through a sieve into a large bowl. Using the back of a wooden spoon, press the celery to release its juice (and flavor). If planning to eat the soup right away, skim the surface and discard any fat. If not, cool the soup completely and chill, covered, in the refrigerator. Before reheating, skim the fat. Serve the chicken soup with torn pieces of the chicken and the carrots, onions, spaghetti, Parmesan and parsley.