Had an email from Mr Scott too pointing out that there's an NCP in Wandsworth, 3 miles away from Battersea Park that you can park at from Thursday 8pm to Sunday 8pm (for example) for £26, then catch the 44 bus between the two (takes about 45 mins, but that's London for you). Seems simple enough.
Here is said car park:
http://www.ncp.co.uk/find-a-car-park/car-parks/wandsworth-garratt-lane/
We have to sympathise with MSVR as there isn't that much room around the area to give us a parking space each, I think we're lucky to get a campsite to be honest!