Put the basil, parsley, red wine vinegar, garlic and some salt and pepper in a medium bowl. Whisk in the olive oil. Set aside.
Brush the eggplant, onion, pepper, squash and zucchini with canola oil on all sides and season with salt and pepper. Brush the cut side of the ciabatta with canola and season with salt and pepper.
Place the zucchini, squash and eggplant on the grill, flesh-side down. Place the onion slices and the pepper on grill. Cover and cook until the cut vegetables are charred, about 4 minutes; then flip and cook, covered, until they are tender, another 2 to 3 minutes. Transfer the cut vegetables to a cutting board. Cover the grill and continue cooking the pepper, turning occasionally, until it is charred all over and tender, about 10 minutes more. Remove the pepper to a bowl, cover tightly with plastic wrap, and let steam for 10 minutes.
Put the tomatoes in a grill basket and grill, covered, shaking the basket occasionally, until charred and softened, about 5 minutes. Grill the ciabatta, cut-side down, until golden and toasty, about 3 minutes. Flip and grill 1 minute more. Transfer to the cutting board with the vegetables.
Seed and chop the bell pepper; remove the peels and coarsely chop the red onions; cut the zucchini, squash, eggplant and bread into 2-inch cubes. Combine the vegetables and bread in a large bowl, season with salt and pepper, and toss with some of the dressing. Mound the salad on a serving platter, top with some of the arugula, and drizzle with the remaining dressing. Scatter the Parmesan over the top, and serve immediately.