Preheat the air fryer to 350°F (175°C).
In a small microwave-safe bowl combine 2 tablespoons butter, the finely minced garlic cloves, and 1/2 teaspoon dried Italian seasoning. Microwave in 15–20 second intervals just until the butter is melted; stir and set the garlic-Italian butter mixture aside.
In a separate small microwave-safe bowl melt the 1 tablespoon of butter (about 10–15 seconds) and set it aside for finishing.
Unroll the crescent roll dough and separate into the 8 triangles on a clean work surface.
In a small bowl combine the shredded mozzarella and shredded Parmesan.
Place a small mound of the cheese mixture on the wide end of each crescent triangle.
Roll each triangle from the wide end toward the point, tucking the seam and folding under the ends to help seal the cheese inside.
Using a brush or the back of a spoon, brush the tops of each rolled crescent with some of the garlic-Italian butter mixture (reserve any remaining mixture).
Arrange the rolls in a single layer in the air fryer basket seam-side down, leaving space between them so they are not touching. Cook in batches if necessary.
Air fry at 350°F for 5–6 minutes, until the rolls are golden brown and the cheese is melted.
Remove the rolls from the air fryer, brush with the 1 tablespoon melted butter, sprinkle with the chopped fresh parsley, and serve warm.