Can you install hardwood over wood parquet?

You could but would you want to? There are some things to consider in this situation. First, the condition of the parquet. Are there loose spots? Will adding hardwood over it present height issues in doorways or adjoining flooring (i.e. tile, carpet, vinyl,etc). Will the baseboard now look to low?
But by removing the parquet your given an opportunity to address the above conditions as well as things like: (1) tightening loose subfloor boards with screws or nailing and (2) replacing damaged subfloor wood if any.
Most likely the parquet is installed on an underlayment. If the parquet is glued to it you also remove the underlayment or you'll have to address the glue that is still stuck to it.
Removing the parquet will entail more work but believe me you'll end up with a better job and a tighter nicer installation.

Now..........if this parquet is somehow glue to concret that's another matter. Also I'd ask the company your considering buying your flooring from the same question you've posted here. Hope this helps.
John

Yes you can but you have to be aware that it will raise your floor level and you'll have to trim doors and door frames.