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Beckenham Place Park has some quite fun downhill runs through the forest, suited to tentative fixed riding as you have to take them quite slowly due to walkers anyway.
Crystal Palace Park isn't so great for it I'd say.
If you can get a bit further, past Bromley there start to be quite a few good bridleways and small woods to explore.
My usual is Spring Park -> over the main road to the little forest where Croydon Rugby Club is listed on Google Maps, then around Layhams Rd, Keston and West Wickham Common there are a few marked paths and bridleways.
Picking up a bike tomorrow to build into a tracklocross set up, anyone in SE got any suitable routes / want to go ride?