Preheat the oven to 375°F (190°C). Lightly grease a 9×13-inch baking dish.
In a large skillet over medium heat, cook the ground beef, chopped onion, and minced garlic until the beef is browned and the onions are softened, about 7-8 minutes. Drain excess fat.
Season the beef mixture with salt, black pepper, and dried thyme. Stir to combine, then remove from heat and set aside.
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, continuing to whisk until the sauce thickens and comes to a simmer, about 5-7 minutes.
Remove the sauce from heat and stir in the shredded cheddar and mozzarella cheeses until melted and smooth.
Arrange half of the sliced potatoes evenly in the bottom of the prepared baking dish.
Spread the cooked ground beef mixture evenly over the potatoes.
Pour half of the cheese sauce over the beef layer.
Layer the remaining sliced potatoes over the cheese sauce.
Pour the remaining cheese sauce evenly over the top layer of potatoes.
Sprinkle grated Parmesan cheese evenly over the top.
Cover the baking dish tightly with aluminum foil and bake for 45 minutes.
Remove the foil and bake uncovered for an additional 25-30 minutes, or until the potatoes are tender and the top is golden brown.
Remove from oven and let the casserole rest for 10 minutes before serving.
Garnish with chopped fresh parsley before serving.