Combine vegetable broth, tomatoes, tomato paste, celery, onions, garlic, carrot, water, oregano, thyme, and pepper in a 3- to 4-quart slow cooker and mix to blend. Cover and cook on Low, 8 to 10 hours.
Stir rice and pesto into the cooker; cover, and let stand until rice is tender, 6 to 8 minutes. Serve soup topped with shredded Parmesan cheese and with crusty bread on the side.
The soup will continue to thicken once the rice is added. More broth or water can be added if desired to thin out the soup.