On average, the truck's weight doesn't change. While the parrots are perched on the floor of the truck, a scale placed under the truck would register the combined weights of the truck and parrots. But even with the parrots flying around inside the truck, the scale would register this same combined weight. That's because the parrots aren't actually weightless—they're able to stay aloft by pushing down hard on the air around them. As required by the physics concept of action and reaction, when a parrot pushes down on the air, the air pushes up on the parrot. The air's upward force on the parrot keeps the parrot from falling while the parrot's downward force on the air causes the air to accelerate downward. This descending air eventually reaches the floor of the truck and pushes downward on it. The floor responds to this downward push by pushing upward on the air and stopping that air's descent. Overall, the parrot's weight is conveyed to the truck floor with the help of the air in between. So even though the parrots are flying, they still push downward on the truck.
Answered by Lou A. Bloomfield of the University of Virginia