Preheat the oven to 350°F.
Spray a 1½-quart baking dish or 9-inch glass pie plate with nonstick spray.
In a food processor, combine cream cheese, Asiago cheese, and garlic. Cover and process until mixture is combined.
Add artichoke hearts, roasted red peppers, mushrooms, and green onions. Cover and pulse just until finely chopped (not pureed).
Transfer the mixture to the prepared pan. Spread evenly in the dish.
Bake the dip for 30 to 40 minutes, or until heated through and bubbly.
Garnish the dip with chopped parsley. Serve warm with pita chips.