This is my shortlist in addition to those listed above;

Brockwell Park and Lido
Nunhead Cemetary.
Chislehurst Caves.
Peckham Rye Park.
Crystal Palace Park Dinosaur Park.
Battersea Park inc Childrens Zoo
The Red House, Bexleyheath.
Thames Barrier.
Belair Park.
Camberwell New Cemetary, Forest Hill Road ( some interesting Romany graves)
Battersea Dogs Home
(kid ye not- its world famous, and Battersea Park still a 25 minute direct train ride away which makes it pretty local for us)

Re Greenwich Park- if you walk east along the Thames Path , beyond the Trafalgar Pub, or west from Maze Hill station, or diagonally east across the park itself and cross over the main road ) you will pass many interesting buildings along the river including an old hospital/care home and some disused industrial plant, and of course some interesting pubs.
I can also recommend getting a boat down the river to the Thames Barrier, which passes many new developments and gives a good flavour (literally) of the industrial past and present of the river.