Paint it. Unless it's that flimsy stuff that likes to bend all over the place. If it's rigid.... Paint it.
Had a car with that problem. I live in Florida so I know what you're going through. Every black piece of plastic down here turns gray after a few years. More like months
Use a 'Matte Black' or something like that. Maybe even a flat black.
Preparation is the key. Remove the pieces if you can, clean them up REALLY well to get that other goop off of it (if it isn't too late already) then prime it and paint it.
You can use a a spray can. It works.
I don't remember exactly what I used.... Plastikote(???).
Found this: http://www.amazon.com/Rust-Oleum-Aut...0_SR160%2C160_
It was so many years ago, I just don't remember. Any paint store should be able to bring you up to date.
If you can't remove the pieces you want to restore, be careful not to get over-spray on everything. Some neighbors have no sense of humor at all