The lease agreement very likely trumps any "rules" the parking co. and HA might try to impose, both by virtue of what it might say and equally what it might not say.
You are being advised by experienced people who do seem to help win cases. If they ask for specific information, it's because it can be important.
Also note: things don't get "escalated to a CCJ".
You might do well to read a few other threads here, after you've answered the question.