Cheesy Zucchini Flatbread

1 10-ounce tube refrigerated pizza dough
3/4 cup garlic-and-herb cheese spread (such as Alouette), divided
3/4 cup finely grated Parmesan cheese, divided
3 tablespoons chopped fresh Italian parsley, divided
1 small red onion
1 7- to 8-inch-long zucchini (yellow or green), cut crosswise into 1/8-inch-thick rounds, divided
Olive oil

Preheat oven to 400°F. Line baking sheet with parchment paper; spray with nonstick spray. Unroll dough onto parchment. Spread half of herb cheese over 1 long half of dough, leaving 1/2-inch plain border. Sprinkle with half of Parmesan and 2 tablespoons parsley. Using parchment as aid, fold plain half of dough over filled half (do not seal edges). Spread remaining herb cheese over top; sprinkle with remaining Parmesan. Remove enough outer layers of onion to yield 2-inch-diameter core; cut into 1/8-inch-thick rounds. Arrange 1 row of zucchini down 1 long side of dough. Arrange onion rounds in row alongside zucchini. Arrange 1 more row of zucchini alongside onion. Brush vegetables with oil; sprinkle with salt and pepper. Bake bread until puffed and deep brown at edges, about 24 minutes. Sprinkle with 1 tablespoon parsley.