Position a rack in the upper third of the oven and preheat to 475 degrees F. Combine 4 tablespoons mayonnaise, the tuna, celery, scallions, parsley and vinegar in a large bowl and mix well. Season with salt and pepper.
Combine the remaining 2 tablespoons mayonnaise with the mustard and horseradish in a small bowl; spread on the bread slices and arrange on a baking sheet. Mound the tuna mixture on top and pat it into an even layer. Top each with 1 cheese slice, tearing it as needed to cover the tuna mixture.
Season the tomato slices with salt; place 2 slices on top of each sandwich, then top each with another cheese slice. Bake until the cheese melts, 5 to 7 minutes. Serve with dill pickles if using.