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 tray and grill 1 minute, or until lightly toasted. Remove from heat and spread with the tuna filing. Place 1 cheese slice over the tuna on each piece of bread, layer with a tomato slice, and top with remaining cheese.
Return layered bread to the grill and cook 3 to 5 minutes until cheese is melted.