Can you solve the problem by keeping at least half the parakeets flying around in the cargo hold the entire time?

There Ain't No Such Thing As A Free Lunch.

The parakeets fly by pushing air downward with their wings, giving a transfer of momentum from the air to the parakeet, lifting the parakeet and pushing the air downwards. The air does not continue going down forever -- it hits the bottom of the cargo hold, transferring the downward momentum imparted by the parakeet's wings to the cargo hold, restoring equilibrium and giving the effect of increasing the weight of the air by the amount that the parakeet's weight was effectively "reduced".

This applies to a closed system such as the cargo hold. In an open system it would be different. For example, if you put a box with an upward-facing propeller on a postage scale, the box would weigh less when the propeller was turning since at least some of the air pushed downwards by the propeller would not hit the weighing surface of the scale.

tanstaafl.