If you are just going to sand and refinish with polyurethene without staining the wood darker:
I would rent a Clarke Floor Maintainer. I have used this machine on over 30 apartments, and the results were all fantastic! You can use 60 grit sanding screens with this machine to clean up the old finish. If your parquet floors are hard wood (Oak, Ash, Maple,etc) you would be ready to apply Polyurethene. If they are soft wood, you should finish with 80 grit screens before applying the finish. (You will have a lot of sweeping/vacuuming before you can apply the finish.)
Go to a tool rental store like Taylor Rentals, or maybe UHaul. Even Home Depot has rentals in my area. They can give you instructions on how to use the machine.
It is heavy work..... Not suitable for someone with a bad back. Be sure to use Safety Glasses, Ear Plugs, and a good respirator!
Also, seal off the area from the rest of the house. You will get sawdust everywhere. You should be able to complete sanding the 2 rooms with a 1/2 day rental.