For sauce, in a food processor or blender combine mayonnaise, the 1 cup olive oil, the tuna, anchovies, lemon juice, and capers. Cover; process or blend until nearly smooth. Transfer to a medium bowl; cover and chill for at least 3 hours or up to 24 hours.
Spoon sauce onto a deep serving platter. Arrange egg halves in sauce. Sprinkle eggs with parsley; sprinkle with salt. Drizzle the 1 tablespoon olive oil over the eggs.