The BPA's Code of Practice is applicable here:
23.6 When you receive a appeal about the issue of a parking charge, you must stop work on processing the charge
immediately. You must not increase the charge until you have replied to the appeal.
Issuing an 'overdue' notice doesn't seem like 'stopping work on processing' to me - you may wish to write to them to express your surprise in receiving an overdue payment notice when they have not yet responded to your appeal. Ask for confirmation that they have received it, and perhaps attach a copy in case they haven't.
Do you have proof of submission?