Boil potatoes in salted water until tender, 12–15 minutes. Drain well and let steam off to dry.
Mash potatoes in a bowl. Stir in parsley, scallions, egg, garlic, breadcrumbs, flour, paprika, salt, and pepper.
Chill the mixture 10 minutes to firm up, then form into 8 patties about 1/2 inch thick.
Heat olive oil in a large skillet over medium heat.
Pan-fry patties in batches until golden and crisp, 3–4 minutes per side.
Transfer to a paper towel–lined plate. Taste and sprinkle with a pinch of salt if needed. Serve hot.