There probably is a way...although I've never attempted it. Any clever and skilled craftsperson should be able to figure out a way to do what you want; it's called 'invention'...

A floating floor is a floor that is not anchored to the subfloor (gravity holds it in place). A floating floor must have an expansion gap around the whole room. That means there will be a nice, big fat gap at each and every wall. The ideal expansion gap is the same thickness as the laminate. The laminate you show is 10mm thick. That means you must have 10mm gap at every single wall and a gap of 10mm on EITHER side of the door jam. Because this is a rental AND the existing floor is VERY WELL installed (baseboards appear to have been done properly) I do NOT suggest you install quarter round or shoe moulding. That means you will LEAVE the gap all the way around. This gap will fill up with crud. And then to add insult to injury you MUST use an underlayment product. Your product of choice has an integrated underpad which is most CERTAINLY plastic stuff. Even if it didn't have it, you would need an underpad to protect the existing floor. The underpad MUST NOT be plastic or vinyl or any product that will trap moisture. That means you need to work with 3mm cork underlay or Quietwalk fiber based product. Either of these products will add $$ cost to your project. But this point is moot because you have a laminate with plastic underlay = absolutely not allowed over an existing wood based floor. Ever. But why no 'plastic'? Because it will cause the moisture underneath the existing floor to catch on the bottom of the plastic and DRIP DOWN onto the existing floor. And you will be left with a fantastically moldy mess that you will be required to pay for. And considering the fact the existing floor is THROUGHOUT the house, you might be required to remove/replace the ENTIRE THING. Because a tenant must bring the home back to the ORIGINAL state. And that state includes an entire house covered in the SAME floor. Right. On to the transition strips that you will WANT to use at the doors...but will not be ALLOWED to use. Again a reducer (what the transition strip is called when moving from a 1/2" step down to 0" inch floor (like concrete or the existing flooring level). Why can't you use it? Because the transition must be ANCHORED. That is to say you would have to affix the transition to the EXISTING floor. IN doing so you risk (100% risk) of damaging the existing floor. That brings us back to leaving the house in the way you found it. And back to the issue of a floating floor over another floating floor - this is bad science. An unstable floor on top of an unstable floor. It would be like building with marbles. The warranty on BOTH floors is void (landlords love to hear these things...you might own them another floor if they find out about another floor sitting on top). In short: Don't do it. You will have one heck of an expensive 'project' on your hands when you move out. A white area rug would do MUCH better and would be MUCH cheaper.

The slab is old. The slab is so old it does NOT have a vapour barrier underneath. The moisture from the ground (all ground has moisture in it...) is evaporating up through the concrete and into the old parquet (photos please). It sounds like the moisture has been trapped by the laminate floor siting on top. Whatever you do, do NOT cover it up and 'ignore it'. You already have mould spores in there. I can guarantee it. Mould needs two things: water and food (aka...wood or wood adhesive). It is ever better if the damp food is kept away from SUNLIGHT (aka...a laminate floor sitting on top making it all nice and warm, dark, damp and cozy). It is possible the former owner KNEW of this issue (perhaps the reason why they COVERED IT UP) and they *might be on the hook for fixing it (check your state and with your realtor...there is often a time limit on these things so PLEASE HURRY)! In Canada, in some places we have 1 year. In other areas we have 6months....and in others just 90 days. Regardless if you are INSIDE your local 'time limit' the previous owner is very likely on the hook for paying for the fix (the proper fix...not the "Hurry up and hide it quick! Before they come back!). Regardless, you will need to remove it all. The laminate, the wood, etc. You will get a good cleaning of the concrete subfloor done. Allow it to dry. Treat it with fungicides, etc and then let it dry again. Now that it is dry, you can throw down a vapour barrier and lay a new floating floor.

Although Kindling would be a timeless tone choice, I like Lunar with your choices that you have shown. If you haven't permanently (put down payments/ordered) set the wall color, cabinets nor countertop selections then I would go with Kindling and maybe explore choices in this order 1. countertops 2. Cabinet color 3. Wall color Floors are such an investment... Wishing you lots of luck during your new project!!!!!