Vehicles need to be taxed and insured if kept on a road �or other public place�.
The car park you are talking about almost certainly falls within the �other public place� description. Who owns it is not an issue; it is who has access to it that counts.
As an analogy, a pub or a restaurant may be owned by an individual or a company, but it is considered a public place for the purposes of a number of laws. For example, public order offences such as drunk and disorderly, threatening behaviour and affray can only be committed in a public place. A pub or restaurant is deemed a public place for those purposes.
So it is with the traffic laws. Public places include car parks to which the public have access, dockside areas to which the public are admitted and driveways to places such as castles and stately homes. This has been tested considerably in court over many years and a large amount of case law exists.
The council are almost certainly correct by telling you that you need tax, and recent changes to the law mean that you need third party insurance as well.