A parquet floor is made up of wood tiles, each of which is inlaid with smaller pieces of wood. The wood on parquet is thinner than that of solid wood floor planks, and the grain runs in several directions, making it a bad idea to fully sand the parquet as you would a regular wood floor. If your parquet floor needs a new finish, consider screening instead of sanding. Screening takes off the top gloss of the finish but won't penetrate the wood.
Set up your rented floor buffer with the heaviest and roughest of your screening disks. Run the machine over the whole parquet floor, starting in one corner and working your way across the room in courses. Grind off the top layer of finish. Vacuum up the dust.
Take off the heavy disk and load the medium one. Screen the floor again, as before, taking off the remaining finish. Vacuum up the dust.
Repeat the process with the finest disk, smoothing out the surface of the floor. Repeat vacuuming.
Apply a coat of polyurethane gloss over the surface of the floor, starting in one corner and brushing it on with a paintbrush. Keep the coat thin and smooth as you apply it, going in courses from the far corner (furthest from the entrance) and working your way out of the room.
Let the polyurethane set overnight. Lightly buff the surface with extra fine sandpaper, using quick, small strokes. Vacuum up the dust.
Brush on a second layer of polyurethane, let it dry, sand it, and apply a third layer. Let the third layer set for two to three days before using the floor.