A search for 'Falmer' on the Carparks4u website (which is generally reliable) finds nothing closer than Lewes or Brighton:
http://www.carparks4u...park_search[q]=falmer
You might want to consider driving into Lewes and parking there. (Lewes Rail Station has 272 spaces, at £4.70 per day). Trains run every 15 minutes to Falmer (where the station is just across the road from Stanmer Park), at 14, 29, 44 and 58 minutes past the hour, with a 7 or 8 minute journey time. (£3.40 return).
Chris