In a small microwave-safe bowl combine 2 tablespoons butter, the finely minced garlic, and 1/2 teaspoon dried Italian seasoning. Microwave until the butter is melted (about 20–30 seconds) and stir to combine.
Unroll the crescent dough and separate into the 8 triangles.
In a small bowl combine the 3/4 cup shredded mozzarella and 1/4 cup shredded Parmesan.
Place a small mound of the cheese mixture on the wide end of each dough triangle.
Roll each triangle from the wide end toward the point, tucking and folding the sides under to help seal the cheese inside. Place the rolls seam-side down in a single layer in the air fryer basket, leaving space between them (cook in batches if necessary).
Brush the tops of the assembled rolls evenly with the garlic–butter mixture from step 1.
Preheat the air fryer to 350°F. Once preheated, air fry the rolls at 350°F for 5–6 minutes, until golden brown and the cheese is melted.
While the rolls cook, melt the 1 tablespoon butter (if not already melted).
Remove the rolls from the air fryer, brush with the 1 tablespoon melted butter, sprinkle with 1 tablespoon chopped fresh parsley, and let cool 1–2 minutes before serving.