In a large skillet over medium heat, heat 1 tablespoon butter and 1 tablespoon oil until butter is melted. Add mahi mahi; season with salt and black pepper. Cook, turning occasionally, until golden on both sides, 4 to 5 minutes per side. Transfer to a plate.
In same skillet over medium-high heat, heat remaining 1 tablespoon oil. Add asparagus and cook, tossing occasionally, until tender, 2 to 4 minutes; season with salt and black pepper. Transfer to plate with fish.
In same skillet over medium heat, melt remaining 2 tablespoons butter. Add garlic and red pepper flakes and cook, stirring occasionally, until fragrant, about 1 minute, then stir in lemon slices, zest, and juice, and parsley. Remove from heat. Return mahi mahi and asparagus to skillet and spoon sauce over.
Top with parsley before serving.