Scrub the potatoes and cut any wider than 2 inches (5 cm) in half. Place in a large stockpot, cover with cold water by 1 inch (2.5 cm), add 1 tablespoon kosher salt, and bring to a boil. Reduce to a steady simmer and cook about 20 minutes, until a fork slides in with no resistance.
Arrange an oven rack in the lower third of the oven and preheat to 450°F (230°C). Line a rimmed baking sheet with parchment. Whisk the olive oil, melted butter, garlic powder, smoked paprika, ¾ teaspoon kosher salt, and black pepper in a small bowl, then spread one-third of the mixture thinly over the parchment.
Drain the potatoes in a colander, return them to the warm pot, and let them sit uncovered for 5 minutes so surface moisture evaporates.
Transfer the potatoes to the oiled baking sheet, spacing them about 2 inches (5 cm) apart. Cover one potato with a small square of parchment and press straight down with a masher or glass until about ½ inch (1.25 cm) thick, stopping as soon as it splits. Repeat with the rest.
Whisk the remaining oil mixture and brush it generously over the tops, working it into the cracks and brushing a little underneath the edges. Roast on the lower rack for 20 minutes, until the bottoms are deep golden and release easily from the parchment.
Slide a thin metal spatula under each potato and flip it over. Sprinkle the Parmesan evenly over the potatoes, if using, and roast for 10 to 12 more minutes, until the edges are dark golden and crisp.
Scatter the chopped parsley over the hot potatoes and serve immediately, while the edges are still crackling.