OP, where is the authority's evidence pack?
You wrote: 'the sign on the junction conveys lawful exception....[to the footway parking prohibition].
What sign, where's your evidence?
Anyway, what's this comment in the decision: 'I find that the sign for resident's permit holders did not indicate that cars were permitted to park on any part of the footway.'
Where did this come from? (the adjudicator is correct, but does she know why I wonder?).
The law:
The traffic signs in the so-called res parking bays are at best contradictory and at worst unlawful. Whatever they are they do NOT permit parking on the footway, it's NOT the prescribed sign. This is Item 4 in the Part 4 sign table here:
https://www.legislation.gov.uk/uksi/2016/362/schedule/4 this may only be used on a WHITE background. The one in the 'areas' here is Item 2, the standard sign.
The sign for unrestricted parking with 4 wheels on the footway is a different beast: Items 13 or 15 in the Part 2 sign table here:
https://www.legislation.gov.uk/uksi/2016/362/schedule/7 these signs stand alone and as can be seen are on a BLUE background.
The variant which is often referred to as regards Item 15 is 'in marked bays'
Also, you cannot have parking bays in a road which is also covered by waiting restrictions(not the one immediately ahead of you because the white zig-zags don't convey a waiting restriction, but the next one which is adjacent to a SYL).
IMO, the ONLY rational analysis based upon the upright traffic signs in situ is that you and they were in a larger area of permitted footway parking, the only difference being that in the areas marked with prescribed parking place markings (which are NOT the markings for footway parking 'in marked bays') parking place restrictions also apply.
For every one council which understands the distinction there must be 100s which do not.
But as you didn't make these arguments, I cannot see a review succeeding unless there's something in the council evidence. Hopefully, it should be possible to put forward an argument to the effect that the adjudicator's decision was irrational having regard to the fact that they'd already determined that the res parking places (presumably by virtue of the traffic signs) did not permit footway parking from which it must follow that as this was endorsed and permitted by the council then these areas and where you were must have been located in a larger area where unconditional footway parking with 4 wheels on the footway is permitted. Where this starts and ends isn't your problem IMO, it's the council's.