Heat oil and garlic in Dutch oven over medium-low heat, stirring occasionally, until garlic is light golden, 3 to 5 minutes. Add carrots, onion, celery, and ¼ teaspoon salt, increase heat to medium, and cook, stirring occasionally, until vegetables are just beginning to brown, 10 to 12 minutes.
Stir in thyme and tomato paste and cook until fragrant, about 30 seconds. Stir in broth and bay leaves, scraping up any browned bits, and bring to simmer. Stir in rice, return to simmer, cover, and cook over medium-low heat until rice is tender, 40 to 50 minutes.
Discard bay leaves. Stir in chicken and spinach and cook over low heat, stirring occasionally, until chicken is cooked through and spinach is wilted, 3 to 5 minutes. Off heat, stir in parsley and season with salt and pepper to taste. Serve.
Ingredient Note: Essential Nutrients
