Heat the oil in a saute pan and gently cook the onion until translucent, about 5 minutes. Add the carrots, green beans and potatoes. Pour in the vegetable water or stock. Season with salt and pepper and add the bay leaf and thyme. Bring to a boil and simmer for 10 minutes. Add the kidney beans and zucchini and continue cooking until all the vegetables are tender, about 10 minutes longer. Stir in the tomatoes.
Ladle the soup into bowls. (Remove the bay leaf and thyme sprig.) Add a spoonful of pesto to each bowl and grate Parmesan over each serving.