Preheat the oven to 425°F. Cover a quarter sheet pan or a heavy-duty 13"x9" baking pan with aluminum foil. Place a piece of parchment paper on top of the foil. Lightly rub 1/2 teaspoon or less of olive oil over the parchment paper.
Slice the red potatoes thinly, but not paper-thin to avoid sticking and burning. Discard the ends. In a mixing bowl, combine the potatoes, olive oil, ½ teaspoon of salt, and the Parmigiano, tossing until well-mixed. Transfer the seasoned potatoes onto the baking sheet, scraping any residual oil from the bowl. Arrange the potatoes in a single layer, slightly overlapping the slices. Aim for one layer of overlapping slices without needing to make neat rows or be overly precise.
Bake the potatoes until tender and turning a light golden brown (they will be somewhat shrunken), about 20-25 minutes. Remove the baking sheet from the oven.
In a small bowl, whisk together the mayonnaise, the mustard, one teaspoon of the lemon zest, the lemon juice, and two teaspoons of the chopped parsley. In a separate small bowl, combine the remaining teaspoon of lemon zest with the remaining two teaspoons of parsley and reserve for garnish.
Cut the salmon into two or three equal-sized pieces. (I prefer squares over strips.) Season them all over with salt and pepper.
Arrange the salmon pieces, spaced equally apart, on top of the potatoes. Using a spoon or brush, coat the tops and sides of each piece of salmon with the mayonnaise mixture.
To serve, garnish each salmon piece with a lemon slice and sprinkle the reserved zest, cheese, and parsley mixture over the fish and potatoes. Use a metal spatula to transfer a portion of the potatoes (cutting through some if necessary) and a piece of salmon to each plate. Serve right away.
