Preheat a grill or grill pan to medium-high. Lightly brush vegetables with 1/4 cup olive oil. Place vegetables on the grill, reduce heat to medium.
Cook the vegetables until nicely marked and slightly softened, except for eggplant. Cook eggplant until easily pierced with a knife. (Alternatively, roast the vegetables at 400 degrees for 15 minutes.)
Preheat oven to 375 degrees. Spray a 13 x 9-inch baking dish with cooking spray.
Spread approximately 1/2 cup marinara sauce over the bottom of the prepared baking dish.
Place zucchini over the sauce to cover the bottom, then top with 1/3 cup of the marinara and sprinkle with approximately 1/2 cup of the Mozzarella, a sprinkling of Parmesan, and a few leaves of torn basil. (You'll repeat the sauce, cheese, and basil after each layer of vegetables).
Cut the yellow and red bell pepper in half widthwise.
Arrange the yellow bell pepper over the first layer of zucchini, sauce and cheese. Top with 1/3 cup sauce, 1/2 cup mozzarella, and a sprinkling of Parmesan.
Repeat with the eggplant, the red bell pepper, and then the yellow squash, adding a layer of sauce, cheese, and basil between each layer of casserole.
Pour remaining marinara sauce over the casserole. Top with remaining Mozzarella and more Parmesan, leaving approximately 1/4 cup for the topping.
Combine remaining 1/4 cup Parmesan, panko, parsley and remaining 2 tablespoons olive oil in a small bowl. Sprinkle over the top of the casserole.
Bake for 45 minutes or until bubbling and topping is lightly browned. Cover lightly with aluminum foil if the topping browns too quickly.