There is parking for a large number of cars at the Palm Gateway station (at the land end of the Palm) and the parking is free. However, as the parking at the Palm Gateway station is so difficult to find I thought I would add a map to show how to get to it.
The parking is also used for people who use the tram and it has become quite busy and full in recent months but multiple levels of the car park are now open.
A description of how to find it is as follows:
As you are on the beach road (Al Sufouh Road) and driving towards the Palm, you need to keep in the right hand lane as if you were actually going to drive on to the Palm. After passing the Nakheel Office on your right the road will split into 2 with the right lane going on to the Palm and the left lane continuing towards the Marina. Keep in the left lane under the overpass but slow down. The entrance to the car park is IMMEDIATELY after the turning on to the Palm. There is a small sign and the road is unmade but you should see it on your right.
And here is a close up view of getting into the car park from Google Earth – this gives you a good idea of how tricky the access is – i.e. no proper road! – it is really a makeshift entrance with a guy sitting by the road.
Here is a picture of the entrance to the parking: