Bring a large pot of salted water to a boil. Cook edamame until tender, about 5 minutes. Drain, reserving 1/2 cup liquid, and rinse under cold water.
In a food processor, puree edamame with tahini, lemon juice, garlic, cumin, reserved cooking liquid, 1 tsp. salt, 1/2 tsp. pepper and olive oil. Pulse in parsley. Taste and season with additional salt and pepper, if desired. Cover and chill for at least 1 hour.
Preheat oven to 400ºF. Split each pita in half. Brush with dressing; sprinkle with Parmesan and salt. Cut each pita half into 8 wedges. Place on an ungreased baking sheet and bake until crisp and golden, about 10 minutes. Watch carefully toward end of baking time so pita chips don't burn. Let cool on baking sheet on a wire rack. Serve hummus with pita chips and crudités, if desired.
You can bake the pita chips up to a day in advance. Keep in an airtight container at room temperature.