I desperately need input from flooring experts because I have received different information from a floor installer and a GC, respectively.

This is for a small rental condo (about 500 sqft) on the second floor in a high rise building. Ceiling/floors are concrete There's no tenant underneath.

The current floor is carpet on top of those what I think are old Bruce veneer ply 9x9 tiles from the 60s, kind of like parquet.

My preference for the new floor is a plank floor (engineered or maybe bamboo), either floating or glued down. This is what the floor contractor recommended, who has done excellent work for me; however, that was done on site-finished HW floors in low-rise apartments or single homes (so nailed down, no concrete floor). He has seen the condition of the floor before the current carpet was installed, i.e. the bare floor where the water damaged tiles had been removed and the remaining parquet tiles, but he hasn't seen all the concrete floor underneath.

A GC, who is very familiar with the building, recommends replacing everything with new 12x12 PARQUET tiles, quoting about 9.00/sqft incl. installation of new subfloor. He strongly advises against plank floor, because installation would be very difficult:

"I know that the concrete floors have HUGE unevenness all around and leveling that concrete floor (raising level of it up to 2” to 3” at certain spots) would take a lot of time. Then, some wood type of subflooring would need to be installed on concrete slab and only then the flooring. Whole floor would rise for maybe 4” !
Any other method of hardwood flooring installation in that condo would not be proper and problems would occur after the installation. NO FLOORING company will do it the right way and you would end up with problems.
That is why parquet was installed initially. Floors do not have to be perfectly straight and you can not see unevenness."

I'm not crazy about installing parquet, but is he right? Obviously, I don't want floor guy no. 1 to run into the problems the GC describes above and then end up with a horrible floor.

So, do I need to stick with parquet floors, or is there another good option? And what questions do I ask/info do I give to floor installer no 1? We'll meet at the condo in a couple of days.

Would it be possible to install engineered wood on top of the existing parquet tiles, provided it's possible to "fill in" the part where the water damaged tiles were removed?

I would recommend that you get a quote for a floating plank floor including the prep work. They should bring a straight-edge and check the floor for flatness. The floor may not need to be level to be acceptable for a hardwood floor installation. If a new parquet floor can be installed with reasonable preparation, then a floating plank floor should be practical, too.

You should never install a parquet over top a parquet floor, especially over concrete.
But like Pete mentions a floating engineered works great. I do them all the time over concrete. Industry standards only refer to flat within a 3/16th in ten foot radius not level!

I'll echo this. I've worked with parquet and this is usually the best option.