Preheat oven to 400°F. Brush a 12-cup muffin tin with olive oil.
Slice potatoes about 1/8-inch thick. Pat dry with paper towels.
Whisk melted butter, cream, garlic, thyme, onion powder, paprika, salt, and pepper in a large bowl.
Add potato slices to the bowl and toss until every slice is coated.
Stir in half of the Parmesan.
Stack coated slices into each muffin cup, filling nearly to the top.
Bake for 25 minutes until the centers are tender.
Top with remaining Parmesan and bake 10 minutes more until edges are crisp and golden.
Rest 5 minutes. Loosen with a knife, lift out, and garnish with parsley.