I think it's definitely worth finding out about refinishing. Parquet is trickier than plank flooring to refinish, because the grain runs in different directions from square to square, but the cost difference might not be much.

As to what stain to use, it depends on your decorating style. I don't usually like light floors, but think it could be very interesting look for this floor. But if you're going for a more traditional look, a medium tone would be good; for a more urban/modern style, a very dark stain would be great.

Parquet is becoming trendy again!

It's worth re-doing, but look into having it "screened", which is less destructive than bringing in a drum sander.

I'd recommend a medium brownish shade, because it will NOT go light without a serious finishing battle, and deep floors show every speck of dirt.

Why keep the parquet if you are going to stain it all the same color? Where do the repairs need to be made? Is there parquet in a closet or a corner that could be used to make repairs?

"neighbor has that type of flooring in her 1907 house"

The age difference is liable to mean they actual flooring is VERY different.

The modern stuff may be veneer so thin it is not possible to do much of anything besides remove and replace (old stuff may be 3/4 in thick).

Have an experienced finisher look at it and see what they propose.

A light screening may be possible and adequate before a refinish

" Replacing will be expensive and I'd rather put the $$$ elsewhere. "

That answers your own question.
Put the money where you want it.

to the OP:
that looks like modern parquet floor tile popular from mid sixties to mid 70's. It came unfinished and can likely be be sanded with a good result. the 1970'2 home my parents built had all parquet tiled floors that were redone after living there for 15 years. In those years, the parquet was considered fancier than strip wood. If it is a newer home, less than 20 years old, you may have the issue that someone mentioned above of a thin veneer on a prefinished product, but this look was not very popular 20 years ago.

Thank you all for the responses. ColumbusGuy1 - the house was built in 1955 and I' m pretty sure the floor was original to the house. After posting my first message I pulled up a couple tiles that were damaged, and the floor is a solid 1'' thick red oak. I rented a commercial sander specially for parquet floors and it took me almost 3 hours to sand down one 10x12 room using the roughest grited paper they had...this stuff is like petrofied wood (which is a good thing, I guess). Picture below. I'm going to research other possible methods to sand down because at this rate, it will take me weeks to get the entire house sanded down (1500+ sq feet).

the floor is much nicer in this picture. It will be well worth the effort.

Talk to the company you rented the sander from, they may have a better quality sand paper. It took 4 days straight with three guys working til 10pm to do my 1400 square foot home, if that helps you at all.

Detroit_burb - Yes that actually helps put things into perspective for me. I guess I was expecting this to be easy! Thanks again.

I'd like to see the finished floor.

HI there, I have a parquet floor and have sanded it once but when I applied my Osmo oil it just went dark again, it is a 1970s house and it is thick good quality parquet so could sand again but just wondered if anyone has had an success keeping it the light colour?