Except the fence does not belong to the OP either. I don't know why a couple people suggested it did. I don't know what the law is if the fence is exactly on the property line, but even if it is on the property line and even if the law favored the OP, the neighbor will be upset if the OP paints it.
In addition, it's almost impossible to paint one side of any fence, except maybe a stockade fence where it is too high to see the top. Either the original color will show through to the OP in places, or the OP will have to paint part that shows to the neighbor. For one thing, people on both sides of a fence see different parts of it depending on whether they are standing or sitting, or loooking out the window of the split level the neighbors live in. There is NO way to paint one side of the fence.
Most fence builders put the fence an inch or two inside the property line. In my case, my fence is probably a half inch inside. I'm going by the notion that the property line is where the gutters of my townhouse end and the gutter of my neighbor begins. But because my downspout was in the way, the fence company put the first post between his downspout and mine (there was just enough room) and the first section of fence comes back to just inside my property line. AFAIK the owner of my neighbor's house at the time didn't object and afaik it's too late for them to assert dominion over the fence now. At any rate, I'd be very angry if the current neighbor decided to paint even his side of that section of my fence. Even if the law were on his side, it's a declaration of war, when what he shoudl do is discuss it with me. How much more so when the fence is probably within the neighbor's property.