Preheat oven to 400 degrees F (200 degrees C). Butter a 9x13-inch baking dish.
Bring Alfredo sauce to a simmer in a saucepan over medium-low heat, about 5 minutes; add garlic, salt, and pepper. Reduce heat to low and keep sauce warm.
Mix Parmigiano-Reggiano cheese and Pecorino cheese together in a bowl.
Arrange 1/4 the potato slices in the bottom of the prepared baking dish, overlapping them slightly. Sprinkle 1/4 cup cheese mixture, 1 tablespoon parsley, and 1/3 the ham over potato layer. Repeat layering 2 more times, ending with a final layer of potatoes and 1/4 cup cheese mixture. Pour Alfredo sauce mixture over and around the potatoes.
Bake in the preheated oven until potatoes are tender, sauce is bubbling, and top is golden brown, 25 to 30 minutes. Let potatoes cool for 15 minutes. Sprinkle remaining 1 tablespoon parsley over potatoes before serving.