Sauté the Vegetables: In a large pot over medium heat, warm the olive oil. Add the carrots, parsnips, celery, leek, and onion, cooking for 4–5 minutes while stirring occasionally until the vegetables begin to soften. Stir in the garlic, salt, and pepper, cooking for an additional minute to release their aroma.
Poach the Chicken: Introduce the thyme, tarragon, bay leaf, chicken breasts, and chicken broth to the pot. Bring the mixture to a boil, then reduce the heat to low and cover. Allow the soup to simmer for 15 minutes, ensuring the chicken is fully cooked.
Shred the Chicken: Remove the chicken with tongs onto a cutting board and shred it using two forks. Return the shredded chicken to the pot and simmer for 1–2 more minutes to meld flavors.
Finish and Serve: Remove the thyme, tarragon, and bay leaf. Stir in the fresh parsley, then garnish with additional parsley and freshly ground black pepper before serving.