Place zucchini cubes in colander; sprinkle with 2 teaspoons of salt and let stand to drain for 30 minutes.
Meanwhile, combine in a blender the Roma tomatoes, stewed tomatoes, tomato paste, and water; blend until smooth.
Pour tomato mixture and chicken broth into a large stockpot. Bring to a boil over medium heat; immediately reduce heat to low; stir in sugar and simmer, stirring occasionally.
Meanwhile, heat olive oil in a skillet over medium heat. Stir in garlic, onion, green pepper, and zucchini, and cook and stir until onion is clear and zucchini is lightly brown, about 10 minutes. Transfer vegetables to blender and pulse a few times, just enough to chop.
Add vegetables to the stockpot; stir in lemon juice, dill, tarragon, and parsley. Simmer, uncovered, for 35 minutes, stirring occasionally. Stir in Parmesan cheese, salt, and pepper to taste. Simmer an additional five minutes. Let stand for ten minutes before serving.