Gravel might work or at least hide the problem. Water standing on the side of the road can cause road issues such as pot holes in the long run. Especially if the park is an area that get below freezing over the winter. My concern with filling all the low spots is their will always have a low spot unless the water is directed to an area away from where you want it or an area that can soak it up (Grass).
A few yard of gravel would be the easiest and cheapest solution. I would recommend 3/4" minus rock. This is everything from rock 3/4" diameter rock down to stone dust. It is sometimes called road mix. It packs down to make a solid surface while still allowing drainage. Locally it runs about $15 a yard delivered with a minimum order of $100. You can spread it out with a shovel and wheel barrel whenever you need some but your maintenance man will hate you. Using a loader is the way I would do it, but our park has a loader backhoe on site.
Now for the more involved answer…
Does the park have a consistent slope to it?
Is the right side of the picture asphalt or gravel?
From the picture it looks like the road has a slight uphill slope going up the road and the parking spot on the right is gravel. If this is the case. I would recommend getting ahold of a tractor with a rear blade and angleing it to make a gutter along the edge of the road so the water can drain away from units and the road. I am not talking about anything big just a path for the water to run down. Think of a little ditch 1-2" deep and whatever the width the rear blade makes it. The water will run down the ditch and the dry out quickly after any rain
This solution is by far much more involved and could possibly be way overkill. If you don’t have easy access to a tractor it could be much more expensive. But it will solve the problem even if you get large amounts of rain. If will protect the road and the road base from pothole caused by water seeping under the road surface and freezing or making the road base turn into mud and not support weight from cars or trucks.
I hope this helps,