White paint has been poured over the keypads and card readers on the machines in Listley Street North Car Park, and the holes to take change have been filled with wood chippings.
The damage is thought to have been done some time overnight on Wednesday, with confused residents first noticing it on Thursday morning.
Both machines in the car park in High Town have been affected, leaving people without a capable mobile phone with no means of paying.
Resident Dawn Dawson was nearly late for a dentist appointment due to the delay and tried to contact Shropshire Council, which runs the car park, to report the issue.
She said: "I tried ringing the mobile number provided to report the vandalism but there was just a pre-recorded message with no facility to leave a message about the problem.
"I downloaded the app and paid for my parking on that."
Shropshire Council and West Mercia Police have been contacted for comment.