Set an oven rack about 6 inches from the heat source and preheat the oven's broiler.
Combine tuna, mayonnaise, celery, onion, parsley, and vinegar in a bowl; season with salt and black pepper.
Place bread on a baking sheet; broil in the preheated oven until lightly toasted, 1 minute. Remove from oven; keep broiler on.
Divide tuna salad among bread slices; place 1 cheese slice on top of each. Layer each with 2 tomato slices; top each with 1 cheese slice.
Broil in the preheated oven until cheese is melted, 3 to 5 minutes.
