Welcome to GardenWeb. I'm not a flooring expect but my guess would be that you cannot clean away the stains. There is a good chance that the parquet tiles are solid wood and could be sanded down to clean with a belt sander, with a protective finish then applied.

There are suggestions on the internet, google ...how to get stains out of original parkay floors....

Different stains take different methods to remove them....is it grease? Ink? water spots? Black metal stains.
Try to figure out the stains and you will be more successful removing them
But likely you will have to refinish.

I'm not sure what the stains are, as we just moved in in June of this year. If I were to guess, I would say old food stains because they are directly under the counters. I inserted a photo of the larger of the two most obvious stains.

Here is a link that might be useful: Large stain

Try using a brush with a degrease cleaner, rinse well, let dry.

Do not pour water on the floor, use squeezed out wet rags for the purpose.