Is it necessary to remove the boards underneath the parquet and put plywood there? Spoke to 2 floor guys and one said absolutelt yes and other said that the old original wood used is much better than anything we can install now (of course they will strengthen the original wood with screws)

Really it depends on the condition of the subfloor. If it is rotted or otherwise damaged then it may need to come out. If not, there's no reason not to leave it.

We originally thought we could use our existing floors (original to the space) but after further inspection most had to come up and it was just a better decision to install a new subfloor. Good luck..hope you can save yours. Leslie

You cannot know the condition of the sub-floor until you remove your parquet, if the floor didn't have leaks then the sub-floor is most like in a good shape, the leaks usually happen around the radiators and that's where it rotten mostly.

Call Carlos Salazar from Carlos Wood Floors (JC Martin Renovation) he's a floor expert. He should be able to give you good advice. http://www.jcmartinrenovation.com/wood-floor-refinishing/

we left our subfloor, but then placed ashfelt strips to level out the various heights and installed a new subfloor + sound insulation (felt) + then florboards on top of this.

considering what the floor was like before its still rock solid all these years later.