The way I believe it works is as follows.
The London Local Authorities and Transport for London Act 2003 made it a specific offence to park in front of a dropped kerb (Section 14 of the Act). But as Ethel says, this only applies in London and the Act was never intended to operate anywhere else.
The Traffic Management Act 2004 deals with, amongst other things, road traffic contraventions subject to civil enforcement. The list of contraventions (Paragraph 4) is long however it does not include parking in front of a dropped kerb. (Outside of London) it does includes offences such as parking on verges, footpaths, areas designated for others (like Hackney cab stands).
But this Act has a 'get-out' worded as follows: -
Power to add further offences
5 (1) The appropriate national authority may by regulations amend paragraph 4 so as to add further offences (but only in so far as they relate to stationary vehicles).
5 (2) [relates to powers to add further offences in London only]
5 (3) Before making regulations amending paragraph 4 the appropriate national authority shall consult�
(a) such representatives of chief officers of police, and
(b) such associations of local authorities (if any),
as the authority considers appropriate.
That is the route by which it may be an offence - if the 'national authority' has made it one.
The full reference to the Act quoted above is Section 73, Schedule 1, Part 1
K