I find yes. The remaining water is really minimal and dries within 1-2 minutes.
You should also keep in mind that it dries better when you pull it from the squeegee than when you push it.

I had a large crack somewhere in the parquet, where the water penetrated and gave small discolourations around the crack, not very disturbing as it still looks natural, but just a change. (Maybe go over it with a dry mop at the end to be on the safe side).

Because of the oil, it depends on the cleaning agent. Detergent dissolves oil and grease. So either add a little/gentle cleaning agent to the tank or re-oil the parquet often.