Heat olive oil in a skillet over medium heat. Add onion, carrots, and celery, and sauté until softened, about 5 minutes. Add garlic, oregano, thyme, and rosemary, and sauté for another minute.
Transfer the sautéed vegetables to a slow cooker. Add diced tomatoes, vegetable broth, kidney beans, white beans, and green beans. Stir to combine.
Cover and cook on low for 2-4 hours.
Thirty minutes before serving, add the pasta, zucchini, and spinach. Cover and cook until the pasta is tender. Season with salt and pepper to taste.
Garnish with grated Parmesan cheese and freshly chopped parsley, and serve hot.