In a large nonstick skillet, heat 2 tablespoons of the oil until shimmering. Season the fish with salt and pepper. Add it to the skillet and cook over moderately high heat, turning once, until golden and cooked through, about 6 minutes. Carefully transfer the fish to plates and keep warm.
Wipe out the skillet and add the remaining oil. Add the shallots and capers and cook over moderate heat until the shallots are softened and the capers are slightly crisp, about 3 minutes. Add the lemon juice and simmer until slightly reduced. Add the butter and whisk until incorporated. Season with salt and pepper and stir in the parsley. Spoon the sauce over the fish and serve right away.