There is a gated entrance to fields and an electricity transformer near a footpath stile. It can be easily found off Park Lane / Green Lane / Pumpkin Hill, on the outskirts of Burnham Beeches Nature Reserve. If this area is full, there are other places to park along Pumpkin Hill road, which is located directly across.
The location shown in the picture is a suitable spot for parking a small campervan, but you may need to adjust to the presence of someone repeatedly shining a bright flashlight into your windows every 5-10 minutes until you leave. We stayed for approximately an hour, but eventually left due to concerns. It is an upscale area and appears to have tight security, although no one directly addressed the situation.
The carpark shuts down at 7pm. We had to park along Dorney Wood road instead. Luckily, we found a spacious spot without any houses nearby.