The party termed Mr Parrikar's continuing as chief minister, and in the process "playing with his health", as akin to a "suicidal tendency".
Addressing a press conference here, Goa Congress chief Girish Chodankar asked Mr Parrikar to demit office and take care of his health.
"We are all aware of his health. Hence, on behalf of his well-wishers, I would like to tell the CM that his health is more than important than the government," Mr Chodankar added.
"The way he is playing with his health, it is like a suicidal tendency."
He said the Congress has provided full cooperation to the CM over his health until now, but it was practically impossible to continue doing so for his entire five-year term.
"How long are we going to sympathise with the chief minister for his medical condition?" Mr Chodankar said.
"It shows a completely irresponsible attitude of the BJP to ask a person who is suffering from an ailment to continue working and go through so much stress," he said.
He added the Congress would fully support a new chief minister, if one is appointed, and would not protest or oppose him just for the sake of it.
The BJP hit back with its state unit president Vinay Tendulkar stating it was none of the Congress' business to advise Mr Parrikar.
"The state government is functioning very well under the leadership of Mr Parrikar and this has baffled the Congress," Mr Tendulkar said.