Scrape off old glue on the loose parquet pieces and in the hole that was created using a file or sandpaper. Make the hole and edges as clean as possible so the pieces fit neatly back in place.
Apply a thin line of construction adhesive along the bottom of each piece and press it into place.
Nail the pieces down using parquet nails. These are fine nails, like finishing nails, but short enough that they only go into the subfloor. They hold the parquet in place as the glue dries and also reinforce the floor. Avoid putting nails in right on the edges of the parquet piece or you may cause it to splinter.