Conservation of linear momentum is a rewording of Newton's first law. Of course, this is just a special case of his second law where the net external force is zero.
When we look at a system of particles (such as Earth+atmosphere+airplane) we are actually using Euler's laws of motion.
Euler's first law states: For a system of particles, S, the inertial acceleration of the center of mass is equal to the sum of all external forces. Inertial acceleration just means that the acceleration is with respect to an inertial frame.
It will be easier to apply Steve's though experiment with the box. So consider this box sitting on a scale with air and a bumble bee. Well, maybe we should pick an insect that can actually fly...
To simply the numbers, zero out the scale before putting the bee in. Then when the bee sits on the bottom, the scale will read the weight of the bee.
If the bee jumps up, the scale will:
1. momentarilly read more than his weight as he pushes off,
2. read zero as he goes through the air (not flapping),
3. read more than his weight as he lands,
4. and finally, after he comes to rest, it will read his weight again.
Now if he used his wings instead of his legs you will have the exact same story. Similarly, if he hovers, the scale will read his weight. The increase in pressure on the bottom will equal his weight. If he hovers too long (thinking like a helicopter), the air will start to circulate in a toroidal fashion and it will become increasingly difficult to hover.
