Place eggplant cubes in a colander over a bowl, sprinkle generously with salt, and let sit for 30 minutes. Rinse and pat completely dry.
Cut all remaining vegetables into ¾-inch (2cm) pieces, keeping each type in separate bowls.
Heat 2 tablespoons olive oil in a large skillet over medium-high heat. Add eggplant in a single layer and cook for 4-5 minutes until golden brown. Transfer to a bowl.
Add 1 tablespoon olive oil to the same skillet. Cook zucchini and squash for 3-4 minutes until lightly browned, then transfer to bowl with eggplant.
Add another tablespoon olive oil to the skillet. Cook bell peppers for 3-4 minutes until softened, then transfer to the bowl with other vegetables.
In a Dutch oven, heat remaining tablespoon olive oil over medium heat. Cook onion for 5-6 minutes until softened, then add garlic and cook for 1 more minute.
Add fresh or canned tomatoes, herbes de Provence, oregano, thyme, bay leaf, ½ teaspoon salt, and ¼ teaspoon black pepper. Simmer for 5-7 minutes until slightly thickened.
Gently fold all sautéed vegetables into the tomato sauce. Bring to a gentle simmer, cover, and cook on low for 20-25 minutes, stirring gently every 8-10 minutes.
Remove bay leaf. Taste and adjust seasoning. Stir in fresh basil and sprinkle with parsley. Let rest for 5-10 minutes before serving.