In many areas of the country, routine parking regulation has been de-criminalised (under the Traffic Management Act 2004) and is now dealt with by council staff, rather than the police or traffic wardens. In other areas the old system still applies. (For example, here in Suffolk, Ipswich Borough Council have opted to take over parking matters, so it's council staff who carry out patrols and who issue 'tickets'. Just up the road, in Stowmarket, the old system applies and it's PCSOs - with 'Traffic' flashes on their shoulders - who hand out 'tickets').
So if you still see traffic wardens (or 'traffic' PCSOs) in your area, it's the police who you need to contact. If you see council staff carrying out patrols then, unsurprisingly, it's the council who you should turn to.
However, even where routine parking matters are now under council control, some types of bad parking can still be dealt with by the police. It's an offence under Regulation 103 of the Road Vehicles (Construction and Use) Regulations 1986 to "cause or permit a vehicle to stand on a road so as to cause an unnecessary obstruction". It would seem likely that the truck driver is in breach of that Regulation. So (even if the council look after parking matters in your are) it might still be worth speaking to your local community constable, who will probably 'have a word with' the driver.
Chris