Using spiralizer, cut zucchini into ⅛-inch-thick noodles, then cut noodles into 2-inch lengths; set aside. (You should have 1 pound of zucchini noodles after spiralizing; reserve extra noodles for another use.)
Pat chicken dry with paper towels. Heat 2 tablespoons oil in Dutch oven over medium-high heat until just smoking. Brown chicken on both sides, 8 to 10 minutes; transfer to plate.
Stir broth, onion, carrots, thyme, salt, and pepper into fat left in pot, scraping up any browned bits. Return chicken and any accumulated juices to pot and bring to simmer. Reduce heat to low, cover, and cook until chicken registers 165 degrees, about 10 minutes.
Discard onion. Transfer chicken to cutting board, let cool slightly, then shred into bite-size pieces using 2 forks; discard skin and bones.
Return soup to boil over medium-high heat. Stir in chicken and zucchini and cook until zucchini is tender, about 1 minute. Season with salt and pepper to taste. Drizzle each portion with 1 tablespoon oil and sprinkle with parsley. Serve.