Add olive oil to a large pot or dutch oven over medium heat. Add onions and stir until softened and turning translucent, about 5 minutes.
Add in garlic, carrots, and celery and stir frequently until fragrant, about 4 minutes.
Add in broth, tomatoes, potatoes, bay leaves, salt, pepper, dried thyme, and dried rosemary. Stir and bring to a boil.
Reduce heat to medium-low, add corn and peas and place the lid on the pot to cook for 25 minutes or until the potatoes are softened. The soup will simmer during this time.
Stir in the parsley, remove the bay leaves and season with more salt if needed. Serve into bowls, top with parmesan cheese, and enjoy with warm crusty bread!