In a joint statement, the Social Democratic Liberal Party, National Federation Party, People's Democratic Party, the Indo-Fijian community-dominated Fiji Labour Party, One Fiji Party, and the Fiji United Freedom Party charged that the elections were not free and fair, Xinhua reported.
The six parties claimed that, according to their polling agents, a number of ballot papers were removed from polling stations without being counted.
The parties said they have notified the country's Electoral Commission about their concerns.
Meanwhile, the multinational observer group, which has been monitoring the 2014 general election process, said Thursday in its provisional report that in its view, the election was credible and the outcome will broadly represent the Fijian voters and the conditions were in place for those people to exercise their voting rights freely.
Led by Australia, Indonesia as well as India, the multinational observer group, invited by the Fijian government, congratulated the Fijian people on taking a key step in the South Pacific island nation's return to democracy.
According to results of the around 400,000 valid votes that have already been tallied, Fiji First had around 60 percent of the total votes, the Social Democratic Liberal Party was sitting on around 27 percent, while the National Federation Party enjoyed around 6 percent of the votes.
None of the rest of the parties or independent candidates secured more than 4 percent of the votes.
Nearly 600,000 voters registered for the election, which was held Wednesday. The turnout was high.
Fiji's constitution provides for a unicameral 50-member parliament, which will be the country's supreme authority. Elections are to be held every four years and every Fijian over the age of 18 entitled to vote.
According to the constitution, after a general election, the member elected to the parliament, who is the leader of a political party which has won more than 50 percent of the total number of seats in the parliament, assumes office as the prime minister.
However, if none of the parties has won more than 50 percent of the total seats in parliament, it would be up to the parliament members to nominate and elect the prime minister with a support rate of more than 50 percent.
Ethnic Indians comprise 37 percent of Fiji's population of nearly 870,000. Most of them are descendants of indentured labourers who were brought from India between 1879 and 1916 to work in the country's sugarcane plantations.