Perhaps this is not a good picture, but I fail to see any attractive features of this graveled spot. ESPECIALLY if it is encouraging neighbors to park in what is essentially part of your front yard. I love my own yard/grass too much to lay gravel on some of it! It's like giving up part of your yard and widening the street instead.

If it were me, I'd remove it all immediately and plant grass. And install a barrier (like the split rail you mentioned) at the edge of the street to keep neighbors off it.

The picture looks to me like the street is gravel and there is no particular part of it that belongs to you. Where is the actual property line ... somewhere within the grassed area? If that's the case, you have no legal right to restrict parking. I understand how people like it when the street parking in front of their house is left for them, but there is no way to command or guaranty it. As a rule, people TEND to park in front of their own house on account of convenience. But when you tempt them with nearby shade, it's going to be an incentive for them to work against your wishes. You could be cute about it an install a sign that says "Parking For Irish Only" ... or Cherokee or whatever works in your situations. Or just talk to them. But caution, neighbor's can become very touchy and might rebel on account of your not actually owning the parking space. Is there a possibility that you can park some vehicle there frequently enough that you can subtly send the message you want to actually use that parking space? They might relinquish or at least use it less. Landscaping -- planting -- a portion of your yard, such as an island around the tree that comes right up to the curb, might tend to indicate that you'd like more control of the space. If you can't find something that's subtle or a friendly way of gaining control of the space, I think you could risk starting a battle. Sometimes those evolve into wars.

We are in a similar situation. No one used to park in that area in front of our house because it was muddy and overgrown with blackberry bushes. A month or two ago we had our driveway asphalted and as part of that project we went ahead and had the road shoulder graded and graveled at our own expense. (Our county will never ever get around do it.) Now it's a popular parking space in front of our house and our driveway has become quite the neighborhood turn around to boot!

All this to say... I feel your pain, but if your gravel area is in the public ROW like mine, that's just the way it is. If someone is using that area as their daily parking space, I might do as Yardvaark suggested and start parking there. To try to get them out of the habit.