That coating may have some colorant in it. The lighter spots are bthe natural color of the wood. Parquet designs are particularly difficult to refinish---as you are finding. Reason being you have to sand with the grain of the wood when sanding to make sure there are no crossgrain scratches. The white gummy stuff was the finish. What you need to do to repair the 'damage' is to Sand each square the same way you did those two. The reason is so when you do the next step, the result will be fairly uniform. Otherwise the next step will cause the reverse of the problem you now have and you will have two different squares. Once all the squares are sanded with the steel wool, use a sanding block(a piece of wood about 6" long and 3" wide) with 400 grit sandpaper and lightly sand the entire top. I stress LIGHTLY---do not put any pressure on the block. All you are doing here is removing the build up finish you left along the lines---you do not want to sand the wood. Then buy a small can of stain(one half pint) at a home improvement/hardware store) There should be one called Red Mahagony. Wear a rubber glove(although the stain will wear off your hand/fingers is two-three days) and use a piece of cloth---old Tshirt/etc. Fold the cloth into a small square, dip one corner into the stain and begin rubbing the cloth/stain on the top---makes no difference how/what direction---just rub fairly quickly. You will quickly see how to control the color by rubbing slower/faster or adding more stain and chaging the speed of rub. Let the result dry for a day. Now the decision. Do you intend on setting anything on the top? Vase/knickknack, etc.? If so, you need to know shellac/lacquer finishes( as applied by DIYers) are susceptable to water rings/clouding if things are allowed to remain long term. If you want to put a doily and artificial flowers, that is less of a problem. But, if you want a heavy statue or fresh flower vase, the finish you choose should not be shellac/varnish. Easiest finish is rattle can lacquer. Simply spray it on in three light coats---allowing 6 hours per coat before recoating. Your investment so far would be one can stain(about $5 and the can of lacquer, about $4) Varnish would be my recommendation. You will need to brush it on--unless you can find a spray can. Just follow the directions. You can make a wiping varnish by adding some thinner(about 40% thinner)---it is applied just like you applied the stain, but you need to put on 6 coats to get the protection of three brushed/sprayed coats. Shellac takes a bit of a learning curve to use, plus the shelf life of shellac is short---using shellac will cost over $15 if you can find a small useable can.

Your home is beautiful. I think it'd look nicer matched color-wise to the parquet. Does it attach to the kitchen as well though? I actually really like the parquet - I have a small landing with parquet but it's oak that lightened over the years was wondering about refreshing it & making it a bit darker. Is yours old original parquet & have you ever done that?

They look to be stained red oak finger parquet. A decent hardwood flooring installer could get a solid wood in there and then stain to match. I would match the species (you are welcome to send away a sample of the wood to a lab that will test for the species = $100 for the results) and then stain to compliment. How long ago were they refinished? Do they need to be refinished in the next 5-7 years? How much wood is left on top? Can they be refinished? If, in the off-chance they CANNOT be refinished, I would IGNORE THEM and then put in wood that you WANT. You will do this KNOWING that the parquet will COME OUT sometime in the next 10 years or so. At that time you will simply bring in MORE of the solid hardwood floor and then have everything sanded/stained all at once. I would get more information about how much more LIFE exists in these parquet floors. They may not have much more left on them. This will dictate how much you TRY to work with them. There's no point in ripping your hair out trying to match them when they need full replacement very soon in the future.

From a distance, and based on blurry pictures, your parquet appears to be a better quality than what Patricia is referring to. Cheap HD parquet usually has a mix of Red and White Oak, has seven slats per 6" x 6" square, is prefinished and has a tongue and groove so it can only be refinished once. A better closeup picture could help, but the wood appears to be White Oak, which would look quite nice if sanded and refinished with bleach, white pickle and non yellowing urethane. The better quality finger parquet is generally five or six slat with each square being 4-3/4" or 5-1/2" square. They are 5/16" thick, which doesn't sound like much, but matches the amount of usable wood on 3/4" solid flooring. If you need any wood for patching, I may have some in stock.