In a small bowl, whisk together the olive oil, red wine vinegar and juice from 1/2 lemon. Season with a dash of salt and pepper. Set aside.
Using a mandolin (or a very sharp knife), cut the persimmons into 1/4-inch thick slices. Next, shave the fennel bulb so they're nearly paper thin but not quite. Roughly mince the Italian parsley; set aside.
To a medium bowl, add the persimmons and a teaspoon or two (you can eyeball this measurement) of the dressing. Rub the persimmons with the dressing so they're evenly coated. Arrange the persimmons on a plate in a circular pattern, overlapping them. To make the persimmon "cones," slice a slit in the persimmons, starting from the center and cutting outward. Take the two cut-ends and overlap them, creating a cone-like shape. Atop the fanned out persimmons, arrange the persimmon cones in a circle.
To the same medium bowl, toss together the shaved fennel, Italian parsley and remaining dressing. Give it a taste and adjust the salt and pepper according to taste. Transfer the shaved fennel to the center of the persimmon "cones." Place the shaved lardo (or prosciutto, if using) on top of the fanned out persimmons.