First off, it's never a good idea to go through Central Park at night. While Central Park is an idyllic tourist attraction during the day, it is technically supposed to be closed late at night and in the early hours of the morning. The vast majority of cases of crime and violence in the park occur at night.
Central London has three major restricted flying zones which drone pilots need to be aware of. R157, R158 & R159. As mentioned above, R157 overlaps a small part of Battersea Heliport's ATZ. The red circle located within R158 represents a danger area around GVS Citigen's gas plant.
The FAA defines the airspace where drones are allowed to fly and generally, as long as they are not within 5 miles of an airport, they can be flown up to 400 feet above the ground and even up to 400 feet above a structure. So if you're not next to an airport, the FAA allows flying next to a skyscraper.