Hello, I've repared some parkers necks in the past. The neck of a PArker guitar (Fly model) are very special. It use carbon. You cold make a good work if you use black nail polish. Try to buy a good black polish. You can use black nitro too. This is my secret:
Try painting by layers. Slowly wait until each layer is dry before adding any more. You have to get the paint to protrude a little from the neck. Once it has dried and the paint stands out use this trick. Take a quality kitchen paper and wipe it a little on removing enamel without acetone. Lightly drier the kitchen blotting paper until the layers go down . If you are careful and have talent you will look great. The best thing is that you can try as many times as you want because it is reversible. Another more professional trick but more difficult way is to use cyanoacrylate. But it is better for a professional to do it. Good luck
P.S The PArker Fly Spreme is the best guitar ever built. I love it.
Albert
This is probably just fine for a small chip repair, but there's quite a bit of paint missing. Unfortunately it sounds like the bond between the carbon fiber and the paint is failing along the entire neck.
Also, nail polish is typically enamel. The finish on a fly is urethane. You won't get a great bond, and it won't harden the way urethane will, so you can't level and polish. It could easily become a sticky mess which will be hard to remove.
As to why the paint is failing, it may be there was a cleaning step that got overlooked after the carbon was applied and baked, or maybe the baking was rushed somehow so there wasn't a full epoxy cure before paint. (Pure speculation on my part here.)
It was nice of you to post, but IMHO, nail polish is a very temporary fix for the OP, and may even complicate a refinish later on. I'd personally avoid it, but glad you're happy with your results.