Heat grill to high. Brush trout on both sides with oil and season with salt and pepper. Grill, skin-side down, until slightly charred and crisp, 2 to 3 minutes. Carefully turn each fish over and continue cooking until just cooked through, 3 to 4 minutes longer. Remove from the grill and top each with 2 tablespoons Lemon-Parsley Butter. Garnish with parsley sprigs and lemon wedges and serve.
Mix all ingredients together in a medium bowl and season with salt and pepper, to taste. Cover and refrigerate for at least 1 hour before using.