If parquet flooring has been installed correctly, all the individual pieces have been glued down. Removing parquet flooring is a huge job requiring heavy tools, muscle power, time, patience, and makes a big mess. Besides, sometimes the underflowing that the parquet has been glued down onto becomes damaged in the process, and will need to be repaired or replaced before any other type of floor covering can be installed.
The easiest way to update parquet flooring is to simply purchase large area rugs in modern colours and patterns.
Leave it alone, it's timeless. Add a rug
You should keep it unless there are serious issues with the flooring that will make the major effort of removing it worthwhile. But to modernise its looks, why not sand the varnish out and apply different stain colors/shades? The beauty of parquet is that you are dealing with consistently-sized squares laid on a perpendicular basis. So you can play with various designs on graph paper or online until you find something that you like. Then it's only a question selecting the contrasting shades, applying the stain as per your pattern, and finishing it off with 2 or 3 coats of varnish. Or use Danish oil of various shades for a similar effect.
Oh, one more thing: avoid the checkerboard pattern; you will tire of it quickly.
I'd keep it, I think it's been coming back into fashion!!