I think you're on the money here. By accident, the OP did not follow the instructions on the website or PCN(there wouldn't be any mention of Equita on a PCN) and has set a hare running. As suggested by others, it's improbable that when the OP wrote to Equita thinking they were dealing with the council they didn't include their PCN number, so all the OP needs to do is to look at their email to Equita.
OP, this is Cumberland Council's relevant web page:
You have 2 opportunities to challenge a Penalty Charge Notice (PCN) depending on what stage it is at.
Contact with Equita, or Newlyns or Capita for that matter, does not arise until the preceding process has been followed and exhausted.
OP, if the PCN was served on 12th Jan, then the earliest you could be served with a NTO is tomorrow(today is day 28). If you want to try and squeeze in informal reps, then contact the council tomorrow and find out your PCN number, or perhaps it is on your email to Equita