The SRB segments vere regularly reused, not sure about the parachutes and the nozzle stearing gear. Still reportedly it was more expensive to reuse the segments (basically big metal tubes) than to build new set if SRBs for each flight, possibly using better techniques (monolithic carbon fibre overwrapped solid motors, like on Ariane 5/6).

It doesn't really matter much because a look at the actual numbers shows that SpaceX charge more than the cost of launching the exact same payload would have cost using the shuttle. Besides the reusability point is disingenuous when talking cost since SpaceX's cost have actually gone up per (re)launch, not down. So yes, it is more complex than quippy internet back and forth suggests.

Here is a video that explains it in decent details if you are interested, but the TL;DR is that SpaceX is more expensive than the shuttle and way more expensive than they said they would be: https://youtu.be/4TxkE_oYrjU

> s far as I know no solid state boosters were ever re-used (how would that work?)

Nitpicking of "reuse" vs "refurbish" aside the SRBs were significantly reused:

> The RSRM was designed to make the most use of recoverable hardware. The majority of metal hardware was recycled through ATK’s Clearfield refurbishment plant in Utah and returned to a flight-qualified conditioned.

https://ntrs.nasa.gov/citations/20120001536

The boosters used for the final mission, STS-135, even included parts from STS-1! https://spaceflightnow.com/shuttle/sts135/fdf/135srbs.pdf