Snap off the hard, woody ends of the asparagus by gently bending it a few inches from the stem end. The asparagus will naturally snap at the proper place. Discard the stems and place the stalks in a large mixing bowl.
Drizzle the asparagus with olive oil, then sprinkle with the salt, pepper, and thyme. Toss to coat.
Preheat your air fryer to 400°F. Arrange half of the asparagus in the air fryer basket in an even layer.
Air fry for 6 to 8 minutes, tossing once halfway through. If your spears are super thin, start checking at 4 minutes to make sure they don't burn. Repeat with remaining asparagus (if you like, you can keep batches warm in the oven). Sprinkle the Parmesan over the top and toss to coat once more. Serve hot, with a squeeze of lemon juice and pinch of red pepper flakes as desired.
Notes
TO STORE: Refrigerate asparagus in an airtight storage container for up to 5 days.
TO REHEAT: Gently rewarm leftovers on a baking sheet in the oven at 350°F or in the microwave.
TO FREEZE: Freeze asparagus in an airtight, freezer-safe storage container for up to 3 months. Let thaw overnight in the refrigerator before reheating.