What can I do with a lot of 4" square parquetry floor pieces?

Coasters. Dicopouge stuff on them...paint them. Glue felt on the bottom so they do not scratch surface.

First, I would save a half a box of them for future repairs. After that, here are a few links with some ideas.

https://www.pinterest.com/jolane40/uses-for-leftover-hard-wood-flooring/

https://www.thisoldhouse.com/ideas/10-uses-wood-flooring-scraps

http://www.countryliving.com/diy-crafts/how-to/g2126/repurposed-wood-hometalk/?

Years ago we had some and husband used them to build what turned out to be my favorite piece of furniture! Using 2x4's, he built a plywood table base that was about 2' wide, 4' long and aprox. 1' high. He covered the plywood with the parquet pieces and added some fat wooden legs we found at Lowe's. I waxed it with floor wax and it was/is wonderful! It started out as a coffee table and has been Kids' play/train table, bedroom bench (all sorts of uses :-) in that capacity! -including sleep platform for visiting grandchild) extra seating in family room, TV stand, etc. It is now almost 40 years old and still so sturdy that my youngest daughter and her family are enjoying it! He also built a matching cube to use as an end table but I don't remember what happened to that. I think one of the kids took it off to college. . .

I agree save some for future repairs then the rest I would use to cover a planter and seal it or them to use either inside or outside of home. Just start with an inexpensive plane square or rectangular shape and apply the pieces by gluing inplace. If you have widow boxes these would look stunning or in home on a shelf or mantle. I'm not sure how many come in a box though.

I would donate them to Habitat for Humanity. They can either use them in a new house or sell them for cash in their Restore.

Save a bunch for repairs - you can never match them later.