In a bowl, mix the tuna, mayonnaise, celery, onion, parsley, and vinegar. Season with salt and pepper.
Place the rye bread slices on a baking sheet, and broil 1 minute in the preheated oven, until lightly toasted. Remove from heat, and spread with the tuna salad. Place 1 cheese slice over the tuna salad on each piece of bread, layer with a tomato slice, and top with remaining cheese slices.
Return layered bread to the preheated oven, and broil 3 to 5 minutes, until cheese is melted.