Step 1In a large heavy pot, heat the oil over medium heat. Add the leeks, garlic, carrots, celery, potatoes, zucchini and parsley. Cook, stirring until tender, 5 to 10 minutes. Add the water, bring to a boil over high heat, reduce the heat and simmer, partially covered, 15 minutes. Season with salt and pepper to taste.
Step 2Simmer 30 more minutes, stirring occasionally. Add the basil and simmer until the vegetables are soft, about 15 minutes more. Ladle 1 cup of soup into a blender and puree; return to the soup and mix well.
Step 3Ladle the soup into warm shallow soup bowls and serve with grated Parmesan.