The stairs are nicely coordinated with the wood floors. Are you changing the floors as well?

The 'parquet' bit isn't an issue. You can sand off the finish (yep....palm sander...while you sit and grind the orange off) and then prime it. You can then add white paint. I would then 'wait and see' if the painted risers do enough to reduce the 'orange' look of the stairs. You can leave the stringer (that's what that side bit is called). After you've lived with it for a few weeks (while giving your back and hands and forearms a rest) you can decide if you want to continue with the sanding/painting.

The ideal situation would be to match the railing with the tread and match the spindles to the riser. But that's a lot of work sanding down spindles and painting.

I think once you paint out the risers you will create a very different looking stair. The orange treads won't look quite so orange. They will look very handsome against the white riser and will look like a better match to the floors.

I suspect the parquet won't look great once painted. Were it mine I'd investigate what's underneath and see if it's more suitable for painting.

I wasn’t keen on painting the parquet because I think it gives it character. But I guess painting the spindles white is out of the question.

This was helpful thank you. Now I know all the different part of the stairs and what should match what. :)

One man's character is another man's hideous. Of course it all depends on personal preference so in that respect there's no real right or wrong. Mine would be to reface the risers and skirt with 1/4" Birch plywood if possible, refinish the treads, paint the spindles, stain the newel post and handrail darker.