The resolution was passed at a meeting of BJP MLAs at a meeting attended by Parrikar, union minister Nitin Gadkari and caretaker Chief Minister Laxmikant Parsekar.
Parrikar left the meeting temporarily when the resolution was passed, BJP MLA Michael Lobo told reporters.
"We want Parrikar as the CM of Goa. We have faxed our resolution to party President Amit Shah," Lobo told reporters.
Parrikar was Chief Minister of Goa before he was elevated as the Defence Minister in November 2014.
Lobo also said that the leaders of the BJP, which emerged as the second largest party in the February 4 election with 13 of the 40 seats seats, were in talks with Goa Forward and Maharashtrawadi Gomantak Party (MGP) as well as independents over the formation of an alliance.
Both the regional parties have won three seats each, while three independent candidates have also been elected.
The BJP needs the support of eight MLAs to reach the magic number of 21 seats in the assembly. The Congress has won 17 seats and is the single largest party.
"We are confident of getting the required numbers and stake claim to form the government on Monday," Lobo said.