Heat 3 tablespoons olive oil in a large Dutch oven or pot over medium-high heat. Add the onion and 1/4 teaspoon salt; cover and cook, stirring occasionally, until tender and golden, about 4 minutes.
Reduce the heat to medium and add the eggplant and garlic. Cover and cook, stirring occasionally, until the eggplant is just golden and starts softening, about 4 minutes. Add the tomatoes; cover and cook, stirring occasionally, until the tomatoes just start breaking down, about 3 minutes.
Add the broth, 1/2 teaspoon salt and a few grinds of pepper. Cover and bring to a simmer, then uncover and cook until the eggplant is tender, about 10 minutes. Add the squash and continue simmering until tender, about 5 more minutes; season with salt and pepper.
Combine the parsley, parmesan, lemon juice and the remaining 2 tablespoons olive oil in a small bowl. Top each bowl of soup with some parsley mixture; serve with bread.