Heat olive oil in a large pot or Dutch oven over medium heat. Add onion, carrots, and celery; cook, stirring occasionally, until softened, about 5 minutes. Add garlic, thyme, and rosemary; cook for 1 minute until fragrant.
Pour in chicken broth and bring to a boil. Add chicken breasts; reduce heat to low, cover, and simmer until chicken is cooked through, about 20 minutes. Transfer chicken to a plate and let cool slightly.
Meanwhile, stir rice into the soup; bring back to a boil, then reduce heat to low, cover, and simmer until rice is tender, about 15-18 minutes.
Shred the chicken into bite-sized pieces using two forks. Return shredded chicken to the pot. Season with salt and pepper to taste. Stir in parsley if using. Serve hot.
Tips
For a creamier soup, stir in 1/4 cup heavy cream or half-and-half at the end.
Store leftovers in an airtight container in the refrigerator for up to 4 days. The rice will absorb liquid; thin with broth when reheating.
To make it gluten-free, ensure the broth is gluten-free.