Put tuna in a bowl. Coarsely chop 2 eggs; add to tuna. Cut remaining eggs in half lengthwise and set aside.
Add celery, capers, and onion to bowl; mix well with a fork, breaking tuna into small pieces. Add 3/4 cup mayonnaise and 1/2 teaspoon pepper and mix. Add more mayonnaise, if desired. Season to taste with salt (be cautious - capers are salty).
Arrange lettuce and celery leaves equally in four wide, shallow bowls. Mound tuna salad on greens, sprinkle with parsley, and dust with paprika if desired. Set an egg half alongside each salad and garnish with tomatoes, olives, and lemon wedges. Season salads to taste with juice from wedges and more salt and pepper.