The first step is to identify the type of plastic. Techniques that work for one type of plastic won't work on others. For example, ABS is common for street bike parts. It can be easily sanded, polished, re-painted if needed. ABS responds well to plastic cleaners and polish. However these techniques will NOT work on polypropylene which is common on dirt bikes and some street bikes. Sanding and polishing polypro will just make it dull and look worse.
You mention the parts are black. I've had good results cleaning up small black textured trim pieces using a "fog coat" of semi-gloss spray paint. Not a full wet coat, but just a very light mist of paint and parts look new again.
Anyway, identify the plastic type, then proceed accordingly.