Reduce the oven to 375ºF. Line a baking sheet with parchment paper. (See Kelly's Note.)
In a shallow baking dish, whisk together the egg whites with the mayonnaise. (The mayo helps the eggs adhere to the asparagus.)
In a separate shallow baking dish, combine the Panko breadcrumbs, chopped parsley and Parmesan cheese.
Dip each asparagus spear in the egg mixture until it's thoroughly coated then dredge it in the breadcrumbs, pressing to secure them to the asparagus. Transfer the asparagus to the prepared baking sheet. Repeat the breading process with the remaining asparagus, spacing them about 1 inch apart on the baking sheet. Lightly spray the asparagus with cooking spray and bake them for 15 to 20 minutes until golden brown and crispy.
Remove the asparagus fries from the oven and serve them immediately with the prepared roasted garlic aioli.