Heat a >5 quart Dutch oven (4.73 liters) over medium-high heat. Add oil and then garlic, onions, carrots, celery, potatoes, and thyme to heated oil with a generous dash of salt. Saute for ~5 minutes.
Add in drained tomatoes, bay leaf, stock, salt and pepper. Cover and bring to a boil and then simmer for another 15 minutes, until vegetables are tender.
While soup is cooking, chop parsley and sundried tomatoes together, forming a hand-chopped pesto.
Stir in beans and leftover pasta and season soup to taste with salt and pepper. Serve warm in bowls and finish with a spoon of pesto and parmesan cheese if desired.