Make the sauce. In a small bowl, stir together the mayonnaise, lemon juice, capers, parsley, and paprika. Cover and refrigerate while you prepare the tuna cakes.
Drain the tuna. Drain the tuna very well, then press it with paper towels or the back of a spoon to remove as much moisture as possible.
Mix the tuna cake mixture. In a large bowl, stir together the tuna, green onions, parsley, breadcrumbs, egg, mayonnaise, lemon juice, salt, and Old Bay seasoning just until combined.
Shape the patties. Scoop the mixture into portions and use your hands to form patties about 1/2 inch thick. Place them on a parchment-lined plate or baking sheet.
Chill the patties. Refrigerate the patties for 10 to 15 minutes to help them firm up and hold together while cooking.
Heat the oil. Add enough oil to lightly coat the bottom of a large skillet. Heat over medium to medium-high heat until shimmering.
Cook the first side. Add the patties to the skillet and cook undisturbed for 3 minutes, until golden brown and crisp on the bottom.
Flip and finish cooking. Carefully flip the patties and cook for 2 to 3 minutes more, until golden brown and heated through.
Serve. Serve warm with the lemon caper sauce, if desired.