I've tried twice to comment with pictures and Houzz keeps wiping out my comment. I don't know if they are going to show up later or if they are gone forever.

Anyway, Tilebar has 5 samples for $5. Have a look there and order some samples to see what works with the wood tones you've got. I will try attaching the photos in a separate post.

But, if you have a new vanity, did it come with a new vanity top? If so, you will want your floor tile to coordinate with the stone on the vanity top. That may be the more important match instead of the parquet which will be outside the door.

These pictures refuse to post so here are some links to more ideas from Tilebar. Sorry for the multiple posts - the pictures just would not cooperate

Tilebar Cascais

Sabbia marble hex

Pergola wood taupe

I'd just go with a large-format floor tile that's failrly plain. There are lots of cement-look gray tiles that are light and work with many finishes.

Twenties Ceramic Floor and Wall Tile, Plain Gray · More Info