New hardwwod floors on top of existing ones

Not sure why removing the parquet floors needs any special permission from the city. Usually people do not want to lose ceiling height especially if the ceiling are only 8 feet. In addition, you need to adjust doors etc if you raise the floor by putting the new floor on top of existing. My best guess is that he is trying to save some cost in demo and by not putting a soundproof layer (can be 2mm thick) below new floors.

Could there also be environmental issues?

30 is right. A lot of parquet has been glued down with asbestos so you'll have to test for it and then remediate. So yes, a lot of people float engineered wood over existing parquet, with a soundproof membrane in between. I've seen a lot of them. Visually you can't tell a floor that's floated to one that's not and no one will notice .75 inch loss of ceiling height. It might feel and sound a bit different though. I've been on installs that have had what feel like some hollow spots with some bounce, perhaps because the floors were not entirely level. In most cases I couldn't tell or barely noticed. You have to replace base molding or put in new quarter round above the floated floor to hold it in place. As 300 said, you have to cut the bottom of the doors but that's pretty easy. If your front door is metal you might have to replace it entirely to keep the fire rating but that's still a lot cheaper then demoing the old floor.

Most contractors/architects never tell the city about removal of old floors and any asbestos.

I had a new hardwood floor put over a 1950's parquet floor - it worked out very well. I lived with it for 12 years before I sold the apartment, and never really had an issue. I did new baseboards to give it a more modern look. Mine was plain old hardwood strip which was stained in place after installation. As Jason pointed out above, there were a few places with a creak but nothing major. If I recall right, the removal of the parquet more than doubled the overall estimate, which is what motivated me to put one floor over another.

As long as the existing parquet flooring is sound you are fine going right over. Your building may or may not require soundproofing. One other thing to consider are the doors. You might have to have them trimmed for the new floor height and more importantly the entry door.

You do have to tell your building that you are installing new floors.

I'm not sure anyone can give a good answer without knowing more specifics. If this was a $200,000 apartment in Flushing where this was most of the work being done I'm not sure the answer is the same as if it were a $2 million unit on the UES where you were putting in a new kitchen, baths, etc.