Their letter is the usual tosh we see so often. They keep referring to CPZ signs
A Permit Parking Zone cannot be part of a Controlled Parking Zone. All a CPZ does is control the restriction hours of single-yellow lines within the zone. So if this PPZ has no supporting traffic order, then PCNs for parking without a permit within it are void. The CPZ entry signs give no hint of a PPZ.
In fairness, they seem to have erected repeater signs as well as the entry signs. Frankly the only way to get a proper decision is to take them to London Tribunals, but you first would have to wait for the Notice to Owner, submit your reps again, but this time enhanced from our advice here. Once rejected again, you then can register an appeal at CC. Of course, you have to risk the full PCN penalty there.