Preheat the oven to 375°F (190°C). Grease a 9×13-inch baking dish with butter.
In a large skillet over medium heat, cook the ground beef, diced onion, and minced garlic until the beef is browned and the onion is softened, about 7-8 minutes. Drain excess fat.
Season the beef mixture with salt, black pepper, and dried thyme. Stir to combine and remove from heat.
In a medium saucepan, melt the butter over medium heat. Whisk in the flour and cook for 1-2 minutes until lightly golden, stirring constantly.
Gradually whisk in the milk and cook, whisking frequently, until the sauce thickens and begins to bubble, about 5-7 minutes.
Remove the sauce from heat and stir in shredded cheddar cheese and mozzarella cheese until melted and smooth.
Arrange half of the sliced potatoes in an even layer in the prepared baking dish.
Spread the cooked ground beef mixture evenly over the potatoes.
Pour half of the cheese sauce over the beef layer.
Arrange the remaining sliced potatoes on top, then pour the remaining cheese sauce over the potatoes.
Sprinkle the grated Parmesan cheese evenly over the top layer.
Cover the baking dish tightly with aluminum foil and bake for 45 minutes.
Remove the foil and bake uncovered for an additional 15 minutes, or until the potatoes are tender and the top is golden brown.
Let the casserole rest for 10 minutes before serving. Garnish with chopped fresh parsley.