Pat chicken dry with paper towels and season with salt and pepper.
Heat oil in medium saucepan over medium-high heat until just smoking. Brown chicken, skin side down, until well browned, about 4 minutes. Transfer to plate.
Add onion, carrots, celery, and garlic to now-empty saucepan and cook over medium heat until softened, about 5 minutes. Stir in flour and cook for 1 minute. Slowly whisk in broth and add bay leaf. Bring to simmer and cook until slightly thickened, about 5 minutes.
Add chicken and any accumulated juices to saucepan and cook until chicken is cooked through, about 12 to 15 minutes. Remove chicken from saucepan and shred into bite-size pieces, discarding skin and bones.
Return soup to simmer, stir in noodles, and cook until noodles are tender, about 5 minutes. Off heat, stir in shredded chicken and parsley. Season with salt and pepper to taste.