It is generally thought that the "star" badge to which you refer was used by the London Borough of Southwark Parks Constabulary.
The other "green ringed" badge you refer to was used by the Wirral (Merseyside) Parks Police. However, it should read "Parks Police" as opposed to "Parks Constabulary"
Both forces have long since been disbanded.
The Liverpool badge that you refer to is a helmet plate used by the Liverpool Parks Police. the cap badge is smaller and has on it the full force title.
