Heat oven to 450° F. On a rimmed baking sheet, toss the potatoes with the oil, ½ teaspoon salt, and ¼ teaspoon pepper. Roast, tossing once, until tender, 30 to 35 minutes.
Meanwhile, season the tilapia with ½ teaspoon salt and ¼ teaspoon pepper. Heat 1½ tablespoons of the butter in a large nonstick skillet over medium-high heat. Cook the tilapia in batches until cooked through, 2 to 3 minutes per side, adding more oil to the pan if necessary. Transfer to a plate and tent with foil to keep warm.
Add the wine to the skillet and cook over medium-high heat until reduced by half, 2 to 3 minutes. Remove from heat and whisk in the remaining 3 tablespoons of butter, plus the capers and parsley. Serve with the tilapia and potatoes.
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