OK, to better be able to help you, we need to know more about your floors. I suspect, but could be wrong, that your floors are the thinner, 5/16" floors that were very popular throughout the first part of the 20th century, especially with intricate inlays, parquetry and feature strips. I've seen many in San Francisco, Oakland, Berkeley and the surrounding bay area. Now if your floors are of the thinner 5/16" type, do NOT drive screws into the boards. They will split. The proper restorative approach is to refasten larger, strip type boards with 1" hardwood flooring brads. These have a larger head than your typical #3 finish nail plus a thicker shaft. You will need to pre-drill to avoid splitting the delicate wood. And split easily it will. If your parquet is comprised of some very small pieces of wood, those should be glued back into place. The original type of adhesive used would NOT be appropriate because it was designed to swell those pieces as it was often water-based. So a solvent based wood glue should be used. I think in this case, a small dab of Liquid Nails would be sufficient. Avoid getting it on the surface of the flooring. Now, IF your floors are the thicker, 3/4" variety, you could screw those boards down if you wanted to. I would pre-drill to avoid splitting. Fixing noising subfloors is different than fixing loose flooring. If you can get to them from underneath, try driving thin, wood shims between the joists and subfloor boards to tighten things up. There is also a device called a "Squeakender",
http://www.squeakender.com/
that will fix noisy subfloors from below