My daughter has birds and I know you have to be careful about putting anything in the air. I think you will have to board them out while you are working on the floor; there is no way you will be able to refinish it without dust.

You should rent a plate sander, I believe is what they call them. A drum sander could destroy the parquet in a blink if you are not expereienced with it.

After you sand it down to clean off the surface, use a latex coating. Be sure to let each coat of it dry longer than recommended so you are sure it is absolutely dry.