Bring a pot of salted water to a boil over medium-high heat. Boil asparagus for 5 minutes. Drain in a colander and let cool until easily handled, about 5 minutes.
Combine asparagus, breadcrumbs, and Parmesan cheese in a large bowl. Knead mixture with your hands until it is the consistency of dough. Form 1 tablespoon of the mixture into a tot. Place on a plate. Repeat with the remaining mixture. Freeze tots for 30 minutes.
Preheat the air fryer to 400 degrees F (200 degrees C).
Spray the basket of the air fryer with nonstick cooking spray. Place tots in the basket and spray the tops with cooking spray. Fry for 10 minutes, shaking halfway through the cook time.