What should I do with an unfinished parquet floor during a two-year rental?
I would find it hard to believe that someone would sand a wood floor but then neglect to put a protective finish on it (unless this is a very rustic building like a barn?). It's possible there's varnish there but it's not a glossy type. An unfinished wood floor seems like a negligent decision by the landlord and a bad investment on their part. I would check with him/her. I don't know where you live or what the landlord/tenant laws are, but in general at the end of your lease you are not required to leave the property in pristine condition: normal "wear and tear" (for which you are not responsible) would be considerably higher for an unfinished floor.

In any event, the best thing you can do for any wood floor (regardless of finish) is to keep it clean. If you let sand and dirt accumulate they will act like sandpaper and wear the surface away much more quickly.