Heat oven to 425 F and grease a large sheet pan with olive oil.
Place potatoes in a pot, cover with cold water by an inch, add 1 tablespoon salt, and bring to a boil.
Cook until a knife slides in with slight resistance, about 15 to 18 minutes, then drain well.
Mix olive oil, garlic, salt, pepper, and smoked paprika in a small bowl.
Spread the drained potatoes on the sheet pan, then press each one with a glass or masher until about 1/2 inch thick.
Brush the tops with the garlic oil, then spoon on the melted butter.
Sprinkle Parmesan over the potatoes.
Bake until edges are crisp and cheese turns golden, about 18 to 22 minutes.
Scatter parsley over the hot potatoes.
Serve warm with lemon wedges if you like.