South London is blessed for Green Space. The closest parks to East Dulwich are Peckham Rye (in Peckham), Dulwich Park (in Dulwich) as well as Camberwell Old Cemetery towards Honor Oak Park. If you venture a little further south, you have Horniman Gardens in Forest Hill (great views) and Crystal Palace Park in Crystal Palace. Sydenham has Alexandra Recreation Ground, Sydenham Wells Park, Mayow Park as well as Sydenham Hill Wood, which is part of the Great North Wood from the medieval times. It has one of the best views of London! Good luck with the move!