Preheat the oven to 425°F (220°C). Line a rimmed baking sheet with parchment or lightly oil it.
Scrub the russet potatoes and pat completely dry; do not peel.
Place each potato between two chopsticks or wooden spoon handles and slice crosswise every 1/8 inch, stopping when the knife hits the guides so the base stays intact.
Rinse the sliced potatoes under cold water to remove surface starch, then pat very dry so they crisp well.
Melt the unsalted butter; reserve 1 tablespoon in a small bowl. Stir the olive oil into the remaining melted butter to make a basting mixture.
Brush the potatoes all over and between the slices with about half of the butter–oil mixture; season evenly with the kosher salt and black pepper.
Roast for 25 minutes until the slices begin to fan open.
Brush the potatoes with the remaining butter–oil mixture and return to the oven for 20 minutes more.
Stir the dry breadcrumbs into the reserved 1 tablespoon melted butter until evenly moistened, then sprinkle over the potatoes, letting some fall between the slices.
Roast 10–15 minutes more, until the edges are deep golden and crisp and the centers are tender when pierced with a skewer. Transfer to a platter and sprinkle with the fresh parsley leaves. Serve hot.