* If you prefer, 4 ounces (2 cups) of egg noodles can be substituted for the spaghetti.
Pat chicken dry with paper towels and sprinkle with 1/4 teaspoon salt and 1/4 teaspoon pepper. Heat oil in Dutch oven over medium-high heat until shimmering. Cook chicken until well browned all over, 8 to 10 minutes.
Add broth, onion, carrot, celery, thyme sprigs, bay leaf, and 1/4 teaspoon salt, scraping up any browned bits. Bring to a boil, cover, and reduce heat to low. Simmer until breasts register 160 degrees and/or thighs register at least 175 degrees, 14 to 17 minutes.
Remove pot from heat; discard thyme sprigs and bay leaf. Transfer chicken to plate and let cool slightly. Using 2 forks, shred chicken into bite-size pieces; discard skin and bones.
Return soup to boil over medium-high heat and add pasta. Cook uncovered until pasta is tender, 9 to 11 minutes, sitrring often. Add chicken and parsley and cook until pasta is tender, 9 to 11 minutes, stirring often. Add chicken and parsley and cook until chicken is warmed through, about 2 minutes. Season with salt and pepper to taste. Serve.