Slice potatoes 1/4 inch thick. Rinse briefly and pat very dry.
Warm a large skillet over medium heat. Add olive oil and butter; let butter foam.
Arrange potatoes in one layer. Season with salt, pepper, and paprika. Add water, cover, and cook 8 minutes.
Uncover and cook until tender and golden, 6 to 8 minutes, flipping occasionally.
Stir in minced garlic and rosemary. Cook 1 minute until fragrant.
Raise heat to medium-high. Toss to crisp edges, 2 minutes.
Off heat, add lemon juice, parsley, and Parmesan. Taste and adjust seasoning.