Preheat a heavy soup pot or Dutch oven over medium heat and add 2 Tbsp olive oil. Add chopped onions and carrots and saute for 6-8 minutes, stirring occasionally until golden.
Add celery, canned tomatoes (with juice), broth, potatoes, green beans, bay leaves, salt, and pepper. Bring it to a boil then reduce heat to a simmer and cook for 25 minutes.
Once the vegetables are tender, add corn, sweet peas, green onion, and parsley. Season with salt to taste and simmer for another 5-8 minutes. Remove from heat and serve warm.