1. Halve potatoes lengthwise. Using a very sharp knife or mandolin, slice potatoes lengthwise into paper-thin rounds.

2. Lay a piece of parchment paper on work surface; arrange potato slices, slightly overlapping, to form a 5-by-7-inch rectangle on parchment; season with salt and pepper.

3. Place one fish fillet on potato rectangle. Using the parchment paper, wrap one side of the potatoes around the fish, pressing to adhere. Top with three parsley leaves. Wrap remaining side of potato rectangle around fish and over parsley, pressing to adhere. Carefully remove parchment paper and discard. Repeat process with remaining potatoes, fish fillet, and parsley.

4. Heat olive oil in a large nonstick skillet over medium-high heat. Pat potatoes dry with a clean towel and add fish, parsley side down; cook, turning, until potatoes are golden and crisp on all sides, 2 to 3 minutes per side. Remove from heat and serve immediately with beurre rouge sauce.