Gingernuts, i totally agree about relocating the event. It is a great day out for a huge number of people but unfortunately it can be pretty miserable for those of us living in the immediate vicinity of the museum with respect to parking and sheer numbers of people. To clarify your point, the website request from the Horniman was instituted after repeated reports of concerns from local residents associations regarding parking (and the abusive behaviour towards residents of some drivers when challenged) on the private roads off Sydenham Rise (and also Sydenham Hill Estate) - a particular problem during the big Museum events. On the private roads off Sydenham Rise we all pay a yearly service charge to Dulwich Estate to maintain the service roads and the communal areas; the roads also serve as access for emergency vehicles and as such need to be kept clear - i think if you were paying for something such as a private road, you might get a bit narked if it was just being used as a public car park, with people turning (often in driveways) all day long, wouldn't you? Of course people can park on the public highway (Sydenham Rise itself) but it is only respectful and polite NOT to park on the private roads - unfortunately there are a lot of people who choose to behave otherwise.
I'm all for such events where people can enjoy the area and have fun, but you are right, it's getting too big now, and i think a move to a bigger park with better access and infrastructure could only be a good thing for the event itself and those of us living around the museum.....