Light a grill. Brush the turnips on both sides with 1 tablespoon of olive oil; season with salt and pepper. Grill the turnips over moderately high heat until tender, 2 minutes per side. Transfer to a platter.
In a skillet, cook the garlic in 2 tablespoons of olive oil over high heat just until sizzling. Reduce the heat to low and cook until the garlic is golden, 2 minutes. Off the heat, stir in the parsley and 1/2 teaspoon of pepper. Spoon the oil over the turnips and top with the lemon zest.