Remove the floor trim from around the edges of the floor with your hammer and pry bar. Only remove the trim that sits directly on the floor (this usually isn't the baseboard, but a thin piece of trim in front of the baseboard). Don't break it. Set it aside.
Set up your floor buffer with a medium screening pad. Run it across the parquet floor in a forward-and-back motion, with the grid pattern of the tiles. Continue until the shine is gone from the surface. Vacuum the dust.
Screen the floor a second time, using your fine screening pad, to get it flat and smooth. Vacuum the floor when complete.
Apply the first coat of polyurethane gloss using a paintbrush. Start at the far end of the room from the entrance and work your way out. Keep the coat thin and smooth. Do the whole floor.
Allow the polyurethane to dry for six hours. Use extra-fine sandpaper to dull the shine, by hand, with quick, light stroke. Vacuum the dust.
Apply a polyurethane coat in the same manner as the first. Let it dry, hand-sand it and vacuum. Brush on a third coat of polyurethane. Let it dry for 24 hours.
Reinstall the floor trim around the edges, using your hammer and trim nails.
Remove the floor trim around the edges of the floor with your hammer and pry bar without breaking it and set it aside. Use your hammer and pry bar to remove the parquet tiles from the floor, hammering the straight edge of the pry bar under the tiles one by one to break the glue bond and pop them out. Scrape up the residual glue with a razor scraper.
Measure each side of the floor and mark the middle of each of the spans. Stretch your chalk snap line from each mark to the mark across from it, so you have two intersecting lines across the exposed underlayment. This will divide the area into four squares that meet in the middle. Spread floor adhesive over the center of the floor with your notched trowel, covering a few square feet in one of the four corners where the squares meet.
Set a parquet tile in the adhesive at the intersection, along the two adjacent lines. Set a second tile off the first, fitting them together by their tongue and groove edging.
Spread more adhesive and continue laying parquet tiles, building out from the middle toward the edges. Use a table saw to cut the tiles for the perimeter edges, near the walls. Make the cuts so the tiles will be 3/8 inches smaller than the space, leaving a gap at the walls that will allow the wood of the tiles to expand with climate changes.
Let the adhesive set for 12 hours. Reinstall the floor trim around the edges, using your hammer and trim nails.