Heat oven to 375°F. Grease a 2-quart baking dish with the butter.
Peel sweet potatoes and slice thinly, about 1/8 inch. Pat dry to remove excess moisture.
Warm cream, milk, and butter in a saucepan over medium heat until steaming.
Whisk in garlic, thyme, salt, pepper, paprika, and flour. Simmer 2–3 minutes, whisking until lightly thickened. Remove from heat.
Arrange half the sweet potatoes in the dish, overlapping slightly. Pour over half the garlic cream and sprinkle with half the cheeses.
Repeat with remaining potatoes, cream, and cheeses to form the top layer.
Cover tightly with foil and bake 25 minutes.
Uncover and bake until the potatoes are tender and the top is golden and bubbly, 20–25 minutes.
Rest 10 minutes before serving. Scatter with parsley and serve warm.