No. The Highway Code is not as categoric as that. What it says is:
"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"
That's not written very well (I think there ought to be a comma after "elsewhere") but it's basically saying that in London parking on the pavement is totally forbidden, unless expressly permitted by signage.
Outside London there is no blanket ban on parking on the pavement, but you should do your best to avoid doing so for the reasons given. Outside London you don't need a sign permitting parking on the pavement to do so - so long as you are not obstructing the pavement or causing any other offence.
The key to the difference between London and elsewhere is in the use of "MUST NOT" in respect of London and "should not" in respect of elsewhere.
PS - I believe there is also since last year a blanket ban on pavement parking in Scotland