Parks Police are constables only within their jurisdiction, ie the parks they police. So, saying that they have the same powers as an HO officer is wrong, in as much as those powers are non-existant anywhere but within their jurisdiction.
Saying "a constable is a constable" isn't the whole story. One has to look at the legislation they were appointed under in order to understand exactly what sort of constable they are (and consequently what jurisdiction they have). Constables can be appointed under the Police Act 1996, the British Transport Commission Act 1949 (BTP), the Ministry of Defence Police Act 1987 (MDP) and so on and so forth, each with differing jurisdictions and minor differences in powers.
Addressing an earlier point, HMC&E (now HM Revenue and Customs) did not get their powers of arrest from PACE as statutory undertakers. PACE only references statutory undertakers with reference to a specific power of search for stolen articles leaving an undertaking. HMRC officers obtain their powers of arrest from the Customs and Excise Management Act 1979 and earlier Revenue Acts prior to this. PACE is only applied to HMRC for functions such as custody, production orders and the like.
As far as I know, no security guards have ever had police powers or anything like them. The fact that guards used to carry batons (in fact truncheons) was purely because there had, up to that point, been no court case to say they couldn't. Once a court decided such practices fell foul of the offensive weapons laws it was stopped. Nothing to do with a removal of (non-existant) police powers or the Labour goverment.
Personally, I am in favour of an American system whereby all police officers have to undergo certain mandated training before they can be appointed. This would weed out all these quasi-police forces appointing (technically) constables under very old legislation. What it would do to the special constabulary is another matter (probably professionalise it). I'd also like to see all police officers be appointed under one Act and have a common set of jurisdictions and powers. But that's another matter.