Boil the baby potatoes in salted water for 12 to 15 minutes until fork-tender, then drain and let cool for 2 minutes.
Preheat the air fryer to 400°F / 200°C for 3 minutes.
Place the potatoes on a cutting board and gently press each one flat with the bottom of a glass or measuring cup until about 1/2 inch thick.
Brush the smashed potatoes with olive oil, then sprinkle with garlic powder, smoked paprika, salt, and black pepper.
Arrange the potatoes in a single layer in the air fryer basket and air fry at 400°F / 200°C for 12 to 15 minutes, flipping halfway, until the edges are golden and crispy.
Sprinkle with Parmesan cheese and fresh parsley, then serve immediately while hot and crispy.