There is no parking lot around or near the Rear Range Lighthouse; however, Palmetto Dunes Resort says that you can park in the grass area beside the road at the lighthouse’s entrance.  You can read their response to a visitor post on TripAdvisor.
We decided to ride our bikes when we visited the lighthouse in Palmetto Dunes. This was the best decision in our case because we wanted to experience Leamington, and riding our bikes allowed us to go at a slower pace and take in all of the sights. You will pass the Arthur Hill Golf Course along the way.