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Add the potatoes, sausage, paprika and sprinkle with salt and pepper. Cook, stirring, about 2 minutes. Stir in the rosemary, and then start pressing the mixture onto the griddle so that the bottom gets brown and crispy and the potatoes stick together. Turn the mixture occasionally to distribute the crispy bits.
Toss the mixture with the parsley and transfer to a serving platter. Drizzle with olive oil and serve hot or at room temperature.