Bring a large pot of salted water to a boil. Add the potatoes, cover and cook over medium-high heat until just tender, about 6 minutes; drain.
Heat the butter and olive oil in a large cast-iron skillet over medium heat; add the potatoes and spread into a single layer. Cook, undisturbed, until they start browning on the bottom, about 5 minutes. Add the onion and sprinkle with the paprika; flip the potatoes and cook 5 more minutes. Stir once, then continue cooking until the potatoes are mostly golden brown, about 5 more minutes.
Meanwhile, roughly chop the garlic, pile the parsley on top and sprinkle with 1 teaspoon salt. Chop and mash together until finely minced. When the potatoes are done, add the garlic mixture to the skillet and toss to coat; cook until heated through, about 2 minutes. Season with salt and pepper.