The parties said there is "no democratic atmosphere" for free and fair local bodies' elections.
Opposition CPI-M and Congress have urged the Tripura State Election Commission to defer the September 30 panchayat polls as there is "no democratic atmosphere" for free and fair local bodies' elections.
Communist Party of India-Marxist (CPI-M) members led by party's West Tripura district secretary Pabitra Kar on Tuesday held a demonstration in front of the State Election Commission office.
The Congress had held similar demonstration on Friday.
The ruling Bharatiya Janata Party has denied the charges.
"There is no law and order problem in Tripura now. CPI-M leaders must see their faces in the mirror about what they did during their 25-year rule," BJP spokesman Mrinal Kanti Deb told the media.
CPI-M Tripura state secretary Bijan Dhar said since the issuance of notification of panchayat elections on September 4, "BJP backed anti-social elements have unleashed a reign of terror all over the state to prevent opposition parties from contesting the elections."
"The situation is not conducive for free and fair polls. So, the process of the panchayat elections must be deferred. All candidates should be given adequate security," said Dhar, also a CPI-M central committee member.
Clarifying that the Left parties would not boycott the elections, the CPI-M leader said BJP workers have attacked several all-party meetings in the BDO (Block Development Office) offices, snatched nomination papers and other documents from a large number of CPI-M candidates.
Tripura Pradesh Congress President Birajit Sinha in a separate press conference raised a similar demand to defer the September 30 panchayat elections.
"Not only the opposition party leaders and members were attacked by BJP workers, the supporters of the junior partner of the ruling party, IPFT, too were attacked," Mr Sinha said.
The by-elections to a record number of 3,386 vacant seats of gram panchayats (3,207 seats), panchayat samitis (161) and zila parishads (18) were necessitated by large-scale resignations of elected representatives of Left parties and death of some representatives.