Well the first page you linked to says that, but this page:
https://www.croydon.gov.uk/parking-streets-and-transport/streets-roads-and-pavements/chns/broad-green-chn-parsons-mead-areasays:
"Local residents are able to apply for a permit for up to three vehicles, the CHN map accessible via the link a little further down this webpage shows the permit zone. Exemptions will be made for some vehicles such as buses, black taxis, local school staff, blue badge holders and designated carers."This is hopelessly vague, but I don't think it says you need to apply for a permit. It's unclear how or if the second sentence is related to the first, and the second sentence doesn't even make grammatical sense ("local school staff, blue badge holders and designated carers" are not vehicles) but I think it's reasonable to argue that a diligent motorist with a blue badge would read this paragraph and come to the conclusion that they are exempt, and they do not have to apply for a permit because those are only for local residents.
If the two sentences are intended to be read as one, then it either says that worthy vehicles/people are exempt from being able to apply for permits if they are local residents, which is absurd, or that local residents can apply for up to 3 permits but won't get them unless they are in some very small categories, which is also absurd, and if you read it that way it includes a provision for local residents who own buses but not local companies who own buses, which is even moe absurd, so I think it's reasonable to argue that the two sentences must be interpreted independently.
It's also sounds as if that you have a failure to consider, but we need to see their response and compare it to your appeal to advise yo on that.
If this was me, I'd be taking this one to adjudication. The council can't publish a web page that says you're exempt then turn round and say 'ha ha, tricked you, we didn't mean it, cough up'.