You MUST NOT park partially or wholly on the pavement in London, and should not do so elsewhere unless signs permit it. Parking on the pavement can obstruct and seriously inconvenience pedestrians, people in wheelchairs or with visual impairments and people with prams or pushchairs.
Many of the rules in the Code are legal requirements, and if you disobey these rules you are committing a criminal offence. You may be fined, given penalty points on your licence or be disqualified from driving. In the most serious cases you may be sent to prison. Such rules are identified by the use of the words ‘MUST/MUST NOT’. In addition, the rule includes an abbreviated reference to the legislation which creates the offence. See an explanation of the abbreviations.
Although failure to comply with the other rules of the Code will not, in itself, cause a person to be prosecuted, The Highway Code may be used in evidence in any court proceedings under the Traffic Acts (see The road user and the law) to establish liability. This includes rules which use advisory wording such as ‘should/should not’ or ‘do/do not’.
I think the specific offence is driving on the pavement, which parking obviously requires, but I could be wrong.
From the highway code:
Signs explicitly permitting it are rare.
https://www.gov.uk/guidance/the-highway-code/introduction