Last week, Mr Parrikar left for US for medical treatment.
In the absence of Mr Parrikar, the state has been left "headless" and no one has the authority to take decisions on key issues, like the impending crisis in the mining industry, the Sena's Goa spokesperson, Rakhi Prabhudesai Naik, said in a statement.
The recent statement of chief of Goa Forward Party, which is a BJP ally, Vijai Sardesai that Goa is facing an emergency-like situation is a true reflection of the state of things here, Ms Naik said.
"In the absence of Chief Minister Manohar Parrikar, the state is left headless, and deserves to be ruled by the President of India," she said.
Ms Naik said the state has been left headless at a time when it is facing closure of the mining industry and there is no one to speak for the interest of people dependent on it.
The Supreme Court had last month quashed the second renewal of iron ore mining leases given to 88 companies in Goa in 2015.
The top court said it was giving time till March 15 to mining lease holders, who have been granted a second renewal in violation of its previous directions, to manage their affairs.
Ms Naik alleged that the people dependent on mining industry are only getting false promises and seeing the "melodrama" of delegations going to Delhi.
"No solution has been provided to their problems and certainly, in the current situation, we can't expect a solution from the cabinet advisory committee which is the least bothered about people," she alleged.
Before leaving Goa, Mr Parrikar had formed a three-member cabinet advisory committee comprising Vijai Sardesai, Sudin Dhavalikar and Francis D'Souza to guide the state administration in his absence.
Ms Naik claimed that the committee has not even held a meeting to discuss the situation that may arise after the mines are closed.
The Shiv Sena said the state needed President's rule so that at least the Centre could take over and provide some relief to people dependent on the mining industry.
The Sena is an ally of the BJP at the Centre and in Maharashtra. It does not have any representation in the Goa Assembly.