Further to TheChair's link, in London a fixed penalty notice can be issued (and they often are) for parking with any part of a car's wheel on the pavement (or kerb). Even just having a quarter-tyre width of a wheel on the kerb is enough to get a 'ticket'.
Outside of London fixed penalty notices can't be issued by civilian enforcement officers for parking on a kerb/path unless the local authority has introduced a relevant bye-law. However it's still a
criminal offence to obstruct a public footpath.
Simon Hoare MP introduced a Private Members' Bill [Pavement Parking (Protection of Vulnerable Pedestrians) Bill] into Parliament in June 2015, seeking to ban pavement parking throughout the country. However, like the vast majority of Private Members' Bills, it failed to make the necessary progress through the Commons and was withdrawn in December of that year.
http://www.bbc.co.uk/news/magazine-34988833