Preheat the oven to 450 degrees . On a rimmed baking sheet, toss the potatoes with 1 tbsp. EVOO; season to taste with salt and pepper. Arrange the potatoes cut side down on the baking sheet and roast, turning halfway through, until tender and golden, 18 to 20 minutes. Let cool.
Meanwhile, in a small bowl, toss the onion with the vinegar; let stand 10 minutes. Using a slotted spoon, transfer the onion to a large bowl. Add the remaining 3 tbsp. EVOO to the vinegar; whisk and season.
Toss the roasted potatoes, spinach, tuna, tomatoes, olives and dressing with the onion. Divide among plates and top with the shaved Parmesan.