That is an interesting question.
It really depends on whether the downwards current of air, which is being pushed downwards by the flapping of the parrot's wings, exerts a downward force on the bottom of the cage.
For example, if the bottom of the cage is made up of a wire mesh, the downwards current of air would almost go right through it so the spring balance would no longer measure much of its weight.
However, if the bottom of the cage is able to capture the downwards current of air (such as a thin piece of wood would), the downwards current of air will exert a downward force on the cage, and thus the spring balance will still pick up most of the parrot's weight.
I would also like to add that if the parrot was in an enclosed cage (like a sealed box). The time-averaged weight of the box (as measured by the spring balance) would still be the same whether the parrot was flying or not.
(published on 04/20/12)