Preheat the oven to 350 degrees F. Drain the tuna, reserving 2 tablespoons oil. Toss the tuna, reserved oil, 1/4 cup parsley, the olives, red onion, the juice of 1/2 lemon, the smoked paprika, a pinch of salt and a few grinds of pepper in a medium bowl. Spread the tuna salad on 4 of the pita halves; top with the cheese and the remaining pita halves.
Heat 2 teaspoons olive oil in a large nonstick skillet over medium-high heat. One at a time, add the sandwiches and cook until lightly toasted, about 2 minutes per side. Transfer to a baking sheet and bake until warmed through, about 5 minutes.
Meanwhile, toss the arugula, tomatoes, cucumbers, the remaining 1/4 cup parsley, 2 tablespoons plus 1 teaspoon olive oil and the juice of the remaining 1/2 lemon in a medium bowl; season with salt and pepper. Cut the tuna melts into wedges. Serve with the arugula salad.