One of the three parties in his coalition government, which was formed just nine month ago, Bright Future announced they were withdrawing from the government due to a "breach of trust" after Bjarni Benediktsson reportedly tried to cover up allegations that his father helped to get a convicted paedophile's "honour restored" - a legal term which allowed him to apply for certain jobs.
A letter of recommendation is among the requirements to apply for the process.
The Bright Future party said in a Facebook post that it had "decided to terminate cooperation with the government".
Minister of Justice Sigridur Andersen, also from the Independence Party, said Thursday night that she had informed Benediktsson of his father's letter in July, Icelandic public broadcaster RUV reported.
The convicted friend is Hjalti Sigurjón Hauksson, who was found to be continually raping his stepdaughter on a daily basis since the age of five.
Magnus Sveinn Helgasson, managing editor of Iceland Magazine, told Sputnik the affair could eventually result in a major shake-up of the political scene in Iceland. "It is all extremely unusual".
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The third member of the coalition, The Reform party, have called for another vote. Together, the three-party coalition holds a slim majority with 32 seats.
The Bright Future party has four seats in Iceland's parliament, the Althingi.
The populist Progressive Party are performing strongly at the moment in the polls as immigration and asylum seekers are seen as major issues in Iceland - so, it could prove extremely interesting if an election is called, he added.
"This is not the first time controversy has been linked to the PM - he was implicated in the Panama Papers [which brought down the previous government, in which he was finance minister]".
The Prime Minister's father Mr Sveinsson apologised for writing the letter. The law was introduced in 1940 and was originally created to restore convicted criminals' right to vote. "I told Hjalti to face his action and to repent", Mr. Sveinsson said.
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