Some suggestions, in the hope that someone from the Horniman, the Jerk Cookout itself, the Council, the local police or whoever else might be relevant, might see them:

Encouraging Public Transport Use/Parking
Yes, everyone should come by bus - but since this is not going to happen until the oil runs out....

1) Couldn't some sort of voucher for something free (eg: one child meal) be organised solely for those who come by public transport? I don't know exactly how this would work in the era of the electronically-read Oyster (as opposed to the easily-presentable bus or train ticket), but there must some way.
2) Quite a lot of people there seemed to have come over from Clapham/Tooting and other points South-West. There is a train that runs round from Victoria, through stations around there, to London Bridge, stopping here. This should be highly publicised.
3) A shuttle-bus could be run on an hourly basis from the station up the hill for those who need it (young families, disabled, elderly etc). Or, it could be run from the Perry Vale car-park to encourage use of that.
4) Or how about a park-and-ride scheme, from somewhere a bit further out, to spread the parking sites?
5) Could Sainsbury's be persuaded to put aside half (not all - understand people need this for shopping) their car park for guest's use for free?
6) Publicity, publicity, publicity - of car park sites which are not residential streets, of public transport routes, of road routes into and out of the area other than Honor Oak road AND a large print request for consideration of local residents in both parking and driving behaviour. Roads which are not suitable for parking or are to be coned off should also be highlighted, to dissaude drivers from going to them in the first place, to help avoid queues of traffic going round and round and jamming up the roads because of it.

Traffic

1) We get warnings of gas/water works and things like Pride or the marathon via temporary yellow AA boards - could we not have some of those put up around Forest Hill S.Circular, Honor Oak and Forest Hill roads for a couple of weeks before the event, warning of heavy traffic and delays expected, so that random through-traffic can avoid the area if possible, and residents know what to expect? (Those of them who missed the banner up along the Horniman Gardens railings for the week or so before this year's event)
2) The stewards for the event were very good about giving those entering the park leaflets with a site map: could they not do the same when people are exiting, with a leaflet reminding people of the available routes out, and again, a reminder to have consideration for local residents and patience about traffic queues (with the aim of avoiding bad-tempered 3-point turns blocking roads and making the traffic worse).