Parrikar had addressed a gathering of BJP workers outside Mahalaxmi Temple on August 2, the day he filed his nomination papers, after seeking the blessings of the deity.
He had also visited Immaculate Conception Church in Panaji before submitting his documents.
In his complaint, Sanjay Sarmalkar has demanded that an FIR be registered against Parrikar, who should be barred from contesting the August 23 by-election from Panaji seat.
He has demanded that the returning officer for the poll or the state's chief electoral officer (CEO) probe the matter, lodge an FIR and disqualify Parrikar from contesting the election.
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When contacted, a senior official from the CEO's office said the complaint would be investigated as per the law and necessary action taken if it was found that the model code of conduct was violated.
Congress has fielded senior leader Girish Chodankar against Parrikar.