Heat a large grill pan over medium-high heat. Brush onion, bell pepper, and eggplant with 1 tablespoon oil. Add onion and bell pepper to pan; cook 3 minutes on each side or until grill marks appear. Remove from pan. Add eggplant to pan; cook 3 minutes on each side or until grill marks appear. Remove from pan; coarsely chop vegetables. Combine vegetables, remaining 1 tablespoon oil, parsley, and salt; toss to combine.
Spread 1/4 cup hummus over each flatbread, leaving a 1/2-inch border around edges. Divide vegetables over each flatbread; top each serving with 2 tablespoons cheese. Roll up wraps, and cut diagonally in half.