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With a sharp knife, cut a thin slice from bottom of cucumber so it sits level. Cut out a section of one end of the cucumber to form the cockpit; discard cutout piece. Scoop out flesh from cut section, forming a 1/4-in. shell. For wings, cut medium carrots in half lengthwise. With a vegetable peeler, shave one end of the carrots into a wedge. On each side of cucumber, cut a horizontal slit 3 in. from plane's nose a fourth of the way from the top and another fourth from the bottom. Insert wedge end of carrots into slits to form wings. For bars between wings, insert four toothpicks diagonally through carrots on each side. With kitchen shears, cut excess toothpick above and below wings. For plane's tail, cut a 1-1/2-in.-long piece from the small carrot; cut a 1/4-in.-thick slice from the piece. Cut slice in half diagonally to form a triangle. Cut a 3/4-in. slit on tail end of plane. Insert carrot triangle for tail. For wheels, cut remaining toothpick in half. Insert one on each side, 2 in. from the front of plane; attach olives. If desired, fill cockpit with dip, add remaining olive for pilot and garnish with parsley. To serve, cut cucumber into slices, starting at the tail end. Remove toothpicks and wings.
