Wood – How to repair staining on a parquetry floor

I hate to be the one to tell you this, but there is really no magic putty or filler for shallow scratches and gouges in hardwood flooring. Even though there are a lot of colored wood fillers on the market, they are not designed for high traffic flooring or nuclear dogs. If it was just a couple of scratches you might be able to sneak by with a filler, but not for a refinishing job in a larger area as you describe. Most engineered flooring has 3/16" to 1/4" of hardwood veneer over a plywood base. If the scratches have not gone through the veneer, then you can sand the scratches smooth, stain to match, and refinish with 3 to 4 coats of urethane. I would recommend an oil based urethane. Apply 3 to 4 thin coats rather than 1 or 2 thick coats. Remember, the veneer is fairly thin and usually can only be sanded and refinished 2 or 3 times at best. Once the scratching penetrates the plywood base, your floor is unrepairable.

My advise would be to protect the floors with area rugs, runners, etc or keep the dog's nails cut short, or better yet, train them to behave in the house.

This is an educated guess only because I can't really see or feel what you are describing. This may be old floor wax that has had something damp left sitting on it. Often in older homes folks would use old fashion floor waxes that will discolor or turn white with age, water, friction or heat. You may first try using a little mineral spirits or a mild cleaning agent with a polishing compound, such as Bar Keepers Friend with a soft cloth on a small spot to see if it will remove the white residue. If in fact this floor has been treated with a wax product, the entire floor would have to be stripped of wax, perhaps buff sanded and restained, then protected with a few coats of Urethane. Don't attempt to apply urethane until you have determined what is causing the white waxy residue. Putting any coating such as urethane over a waxed floor will bubble, peel, and create a mess to clean up. another test you could do would be to gently scrape a very small section of the white stain with a putty knife. If the material comes off easily it may be wax. it may be old varnish, but that would not feel waxy, but not out of the question. Let us know what you find, and we can go from there. Good Luck