There is another alternative for screwing down the subfloor in the event that you do not have access to the floor joists from below.
There are special floor screws that can be used to screw into the floor from the floor surface but which break off below the floor surface so that there is no exposed head showing from the surface. You can get them here:
http://www.mcfeelys.com/product.asp?productID=SNM-3320
They are not exactly cheap but they do work quite well. Of course, if you have loose parquet you could remove the parquet and just use regular screws to resecure the subfloor to the joists. Relay the parquet over the subfloor to cover up the screw heads.