I think the scale of the tile makes the parquet look tiny and busy.
Update parquet flooring.
The first step will be to identify which all pieces are damaged and requires replacement.
It could go a long way to making the floors look more cohesive.
You can purchase a ready tinted sealer in various colours or purchase a clear sealer and mix in your own stain.
Dark wood floors always create a sense of luxury in a space and your idea to add area rugs is great for injecting interest and contrast.
A foolproof way to bring parquet floors up to date is to strip and stain them a rich dark walnut.
Would you consider sanding it down and painting it.
Of the floors are made or parquet and require and update they can be done easily without much trouble.
You may be able to have he floor refinished to a lignter shade but it will be expensive and it will still be a parquet floor next to tile.
Stain and seal parquet flooring to ensure that your floor is looking good for as long as possible simply wipe it regularly with a soft damp mop.
Most people nowadays are leaning more toward plank hardwood.
If only a few of the pieces are damaged pry them out and replace them with a new tile.