As far as I know, everything that is chrome or very reflective will get it's reflection directly from the environment around the car and not from the interior itself or the car or other cars. In other words, no self environmentel mapping. Normally you would expect the interior to reflect in those parts in question, but as also the cars don't reflect on each other and also parts of a car doesn't reflect on ther parts on the same car, this also seems to be the case with chrome parts in the interior.
I can remember that this behavior can be also seen in that open top Corvette tuning concept car. A pillar is made of/using chrome.
So it's rather a design flaw from PD, then a glitch, a compromise because there is not enough system resources to get real self environmental mapping running. This technique currently demands high end workstations and even then running in real time is a big deal.
They could change this, if those parts are not using the common mapping technique for all reflection, but just a simple pre programmed texture to be mapped onto the surface, but then you would need to compromise the ability to reflect the environment in real time for these parts and you need to run one more shader at a time. I think we can live with what we have now, but real time self environmental mapping would be mega, super, realistic.