Rinse and halve the potatoes. Bring a large pot of salted water to a boil.
Parboil the potatoes until just fork-tender, 8–10 minutes.
Drain well and let the potatoes steam-dry in the colander for 2 minutes.
Melt the butter with the olive oil in a large skillet over medium heat.
Add potatoes cut side down. Sear until deeply golden, 6–8 minutes, flipping as needed.
Stir in garlic and paprika. Cook 30–60 seconds until fragrant, stirring to coat.
Season with salt and pepper. Toss with lemon juice and parsley, then finish with Parmesan if using. Serve hot.