Combine the drained tuna, mayonnaise, Dijon, celery, red onion, parsley, lemon juice, salt, and pepper. Stir gently until creamy but still textured.
Butter one side of every bread slice. Place four slices butter-side down, then layer each with cheddar, tuna filling, tomato, another slice of cheddar, and the remaining bread with the buttered side facing out.
Heat a large skillet over medium-low to medium heat. Cook the sandwiches for 3 to 4 minutes, pressing lightly, until the bottoms are deeply golden.
Flip carefully and cook for 2 to 4 minutes more, until the second side is crisp and the cheese is fully melted.
Rest the sandwiches for 1 minute, then slice diagonally and serve immediately.