The council said all on and off-street car parks across the district would be free until 8am on Wednesday, April 15.
The situation will then be reviewed following further government guidance.
However, the local authority is repeating government instructions to the public to travel only when absolutely necessary.
Council leader Denise Jeffery said: "We want to do all we can to support those working hard in the fight against coronavirus.
“We are extremely grateful for the continued support and dedication shown by key workers in this crisis and we want them to be able to continue their vital work without having to worry about parking fees.
“We hope free parking will also help reduce reliance on public transport and support people in following social distancing advice.
"It’s more important than ever that we all do what we can to help fight this pandemic by following the latest guidance."
Local Democracy Reporting Service