Place potatoes into a large pot and cover with water; add salt and bring to a boil. Reduce heat to medium-low and simmer until just beginning to soften, 15 to 20 minutes; drain.
Melt butter in a saucepan over medium heat; add flour and whisk until the mixture thickens, 1 to 2 minutes. Pour milk and heavy cream into the saucepan; cook and stir until smooth, 2 to 3 minutes. Melt Cheddar and Asiago cheese into the butter mixture; cook while stirring continually until the mixture is smooth and creamy, 1 to 2 minutes. Season the sauce with salt and pepper.
Preheat oven to 400 degrees F (200 degrees C). Butter a 9x13-inch casserole dish.
Layer about half the sliced potatoes into the bottom of the baking dish. Top the potatoes with sweet onion slices and green onions; pour about 1/3 of the cheese sauce over the onions. Arrange the remaining potato slices into the dish; pour remaining cheese sauce over the potatoes. Sprinkle Parmesan cheese and crushed crackers over the top.
Bake in the preheated oven until hot in the center and beginning to brown at the edge, about 30 minutes. Sprinkle parsley over the top of the dish to serve.
