The suspension is designed to free up parking for those workers who need to access their workplace and to help people who need to pick up food and medication for vulnerable residents and themselves.
In line with Government and public health guidelines, public toilets are closed until further notice.
The free parking applies to all council-run multi-storeys and bay parking across Sunderland - nearly 2,500 spaces altogether.
Sunderland City Council's Cabinet Member for Environment and Transport, Councillor Amy Wilson said: "The City Council continues to do as much as it can to support our residents and businesses.
"We are trusting members of the public to park appropriately and considerately in order that we can all access the services that are vital to us all in the current circumstances.
"It does not mean that drivers should disobey standard parking regulations or the Highway Code. Inappropriate or inconsiderate parking could still result in the need for enforcement action to be taken to protect services and other highways users."
There are more than 1,500 council car parking spaces in the city centre; with nearly 500 spaces at St Mary's multi-storey and more than 650 at Sunniside.