Murray said she had heard of workers getting fines while attending patients and having to walk long distances with heavy equipment.
“Healthcare workers should not be punished for delivering care in our communities,” she said.
“I hope councillors across the chamber will back this motion on Thursday.”
Murray said more than 440 residents had signed a petition backing the scheme, with nurses and healthcare staff describing the daily struggle of parking while delivering essential care in the community.
The motion, which will be proposed by county councillor Mel Allcott, stated the city council “and other district councils that manage local parking enforcement on behalf of the county council have systems capable of issuing healthcare permits”.
The councillors said an “essential attendance” permit scheme available to residents with care needs was “not designed for NHS staff providing mobile care and is not fit for urgent or time-critical healthcare visits”.
“Residents often rely on others to obtain these permits on their behalf, and the scheme does not support emergency responses or end-of-life care at home.
“The absence of a healthcare permit scheme hinders healthcare delivery and places unnecessary stress and financial burden on key NHS staff.”