Parkerizing is an excellent gun finish as long as you keep it soaked with a rust preventing lube.
US firearms have been being parkerized since WWI and stand up very well.
However, it only works if you apply a rust preventing lube so it can soak into the parkerized coating and establish a barrier to moisture.
It may look a little ratty with patches of missing paint, but as long as you keep a thin coat of lube on it, it'll be good to go.
For better protection and just appearance, you may want to coat it at some point.
Until you decide what you want, you can leave it as-is with a thin coat of a good rust preventing lube like CLP Breakfree, or you can apply a simple paint finish.
Many people have just bought a can of hardware store Krylon paint and sprayed it on the outside metal.
All you need to do with this is use something to degrease the metal before spraying so the paint will adhere.
Since even alcohol seems to strip the paint, you can go ahead and use a solvent to strip it, or just use a liquid soap cleaner to degrease the metal.
Most any kitchen spray cleaner will do a good job, then after rinsing and drying with a hair dryer, just spray on a coat of paint.
After the paint fully hardens and cures, wipe it down with CLP.
Most paints are easily removed when you decide to go with a better firearms coating.