The video message was played at a BJP workers' meeting addressed by party president Amit Shah.
Mr Parrikar is undergoing treatment for a pancreatic ailment at a hospital in the US, where he was admitted in the first week of March after being treated at Mumbai's Lilavati Hospital.
In the absence of the chief minister, a three-member advisory committee is looking after the day-to-day affairs of the government. The three-member panel formed by Mr Parrikar comprises ministers Sudin Dhavalikar (MGP), Francis D'Souza (BJP) and Vijai Sardesai (Goa Forward Party, another ruling coalition partner).
The Congress legislators and office-bearers staged a demonstration in Panaji at the memorial of Goa's first chief minister, Dayanand Bandodkar, in support of their demand.
Goa Congress chief Girish Chodankar said, "We have petitioned everyone, including the governor, seeking the appointment of a full-time chief minister in the absence of Mr Parrikar, but nothing seems to be happening."
Today, the party workers and legislators assembled at the memorial of Dayanand Bandodkar and "prayed" that the ruling dispensation led by the BJP appoint a full-time chief minister, he said.
In April, BJP leader Nilesh Cabral had said the condition of Mr Parrikar was improving, but he would travel to his home state only after doctors allow him.
With inputs from PTI