London's eight Royal Parks — Hyde Park, Regent's Park, Richmond Park, Bushy Park, St James's Park, Green Park, Greenwich Park and Kensington Gardens — generally allow dogs off the lead across most open grass areas, provided the dog is genuinely under control and doesn't disturb wildlife or other visitors.
The one firm, well-publicised exception is deer birthing season at Richmond Park and Bushy Park, both of which hold wild deer herds. During this period, dogs on leads is made compulsory across the parks, with signage and patrols enforcing it, since does giving birth and defending fawns are easily provoked by an off-lead dog approaching too closely.
Beyond the deer-season rule, specific zones within individual parks — formal gardens, certain protected wildlife areas, sports pitches during matches — can carry their own lead requirements or dog exclusions, posted on local signage. It's worth checking a specific park's current rules if your visit is timed around spring, when both the deer season and ground-nesting bird activity overlap.