The letter makes it sound like that because it's free to park, they don't need a Parking 'P' sign. Is that true?
Free for what class of vehicles, motorcycles, HGVs, buses, taxis etc?
It's nonsense and they either know and are being deceitful or don't know and are being stupid. Either is not acceptable for a public body empowered to penalise motorists.
Carry on IMO.
There's another current thread where the authority adopted the General Melchett approach(Blackadder Goes Forth) i.e. 'If nothing else works, a total pig-headed unwillingness to look facts in the face will see us [the authority] through' only to cancel the PCN once the owner on our advice had registered their appeal with the adjudicator at which point the authority, as with all bullies who suddenly realise that they're no longer judges in their own cause, cancelled the PCN.
But when they do in your case don't expect an apology.