You need to wait for the next document which is called the Order for Recovery, this will allow you to file a witness statement with the Traffic Enforcement Centre. One of the statutory grounds is that the recipient of the Order for Recovery "
has paid the penalty charge to which the charge certificate relates", see regulation 19 of
The Road User Charging Schemes (Penalty Charges, Adjudication and Enforcement) (England) Regulations 2013.
To avoid any risks with letters being lost in the post, it's sensible to check the amount due on the council website starting around 20 July, and then at least once a week. Once the Order for Recovery has been issued, the charge will go up by another £9 to £138.
At that point you don't need to wait for the Order for Recovery to come in the post, you can download
form TE9, fill it out on your computer (no need to print, they accept a typed signature) and then email it to tec@justice.gov.uk with the PCN number in the subject line (note: when filling out the form, the "
Applicant" is Bristol City Council, not you).
Once that's done the Order for Recovery and the Charge Certificate will be cancelled and case will be referred to the Traffic Penalty Tribunal with just £69 outstanding, we can then look at grounds to get the PCN cancelled completely and hopefully you'll get a refund of the £60 you've already paid.
One obvious ground is the premium rate telephone number, which has won before (see
Paul Bateman v Derbyshire County Council (DJ00037-2209, 10 November 2022))