How to fix a black painted grille?

I thought the 67 barracuda grills were all metal?
I have 68s so they are the plastic finned inserts.
They can be pretty fragile from age. The urethane may make the plastic more resistant to breaking. Possibly glass bead them lightly and repaint since most weren't chromed anyway.
As for the chrome plated diecast & aluminum parts aircraft stripper would take care of the blackout paintjob. Brush on and cover with saran wrap so it has time to work before drying.
It may take a few attempts depending on how stubborn the paint is to soften up enough to wash off.
Some OOOO fine steel wool should get the diecast chrome parts final cleaned.
You will need to do a thorough job with the stripper for the next step to work on the anodized aluminum
100% lye liquid drain cleaner makes short work of stripping the anodizing from the aluminum parts. This is a soak and rinse process. Not really much scrubbing needed to get them stripped off. I watched them very close not to leave them submerged too long as it can eat aluminum. Any paint left on will protect the anodizing from the lye.
I then used very fine wet or dry sandpaper in a sink of dish water as a last step before polishing. I believe it was 1000 to 1200 grit. This also removes any of the last traces of anodizing which must come off before moving forward to the next step.
To polish I used Mothers mag wheel paste and their red foam ball. It has a light touch so not to dent or damage the thin aluminum. They end up looking like a mirror when done. This shine will dull and have to be repolished every so often. They slowly oxidize without the protective anodizing over time.