Whisk the flour and baking powder in a medium bowl and make a well in the center. Gradually whisk the beer into the well to make a smooth, thin batter. Cover and refrigerate at least 1 hour. (You can make the batter up to 12 hours in advance; it gets better the longer it sits.)
Preheat the oven to 400. Cover the potatoes with water in a medium pot and add a pinch of salt. Bring to a boil over medium-high heat and cook until fork-tender, about 20 minutes. Drain and pat dry. Toss the potatoes with the melted butter and spread on a baking sheet. Slightly flatten each potato with a spatula (do not mash completely) and season with salt and pepper. Roast in the oven until crisp and golden, 15 to 20 minutes.
Meanwhile, heat 2 inches of vegetable oil in a heavy-bottomed pot over medium heat until a deep-fry thermometer registers 375 degrees. Season the fish all over with salt and pepper. Working in batches, dip the fish into the batter, let the excess drip off and lower into the hot oil. Fry until evenly brown, turning with tongs, about 2 minutes per side. Transfer to a paper-towel-lined plate to drain.
Transfer the potatoes to bowls or plates along with the fried fish. Sprinkle with the parsley and serve with malt vinegar, if desired.