Yes we had 'Parkies' in London too. I did find some info on English heritage site, which suggested that during the period my g. grandad was working there were, in some places, distinctions between Park Keepers and Park Constables, Sargents and Inspectors. Sometimes the 'police' type titles were given to people employed specifically on 'security' type duties and at some times and in some places they were given powers similar to 'specials'. I can't find any examples of uniform they might have worn in London at this time, when I was young in the 60's 'Parkies' uniforms were brown and had flat caps and did not look at all like police uniforms.
Regards Susan