Place tuna, scallions, salt, pepper, 1 cup of the panko, and 1 cup of the mayonnaise in a medium bowl. Stir until combined, and form into 4 (¾-in.-thick) patties. Place remaining ½ cup panko in a shallow dish. Dredge patties in panko, pressing lightly to adhere.
Heat oil in a large nonstick skillet over medium. Add patties; cook, turning occasionally, until golden-brown on both sides, 7 to 8 minutes. Remove from heat, and drain on paper towels.
Whisk together parsley, relish, mustard, hot pepper sauce, and remaining ⅔ cup mayonnaise in a small bowl. Serve with tuna cakes.