If you can post the lease/contract/AST for your flat that would be helpful, but in general this is primary and unless it has been formally modified to require a permit to be displayed, putting up a sign has no effect on your legal position.
These companies will deny this and reject any appeal you make, so you sit tight and ignore letters from debt collectors, come back when you get a Letter of Claim which you will simply deny, then defend the county court claim. This is unlikely ever to see court because your legal position is strong, the money grabbing parking companies just want to frighten you into paying or to ignore the court claim which will result in a default judgment against you.
Once they give in, you need to write to the management company and the parking company along the lines that you are not required to display a “permit” but that you will do so as a matter of convenience, but will again defend any future action against you for not doing so.
The approach you already took is the correct one, so I suggest just doing it again, presumably it’s simple to cut/paste the words you used. If you end up with 3 identical court claims, it’s no big deal.