Saeed Mamuzini, of the Kurdish Democratic Party, said the extremists destroyed the Mother of Aid Church in Mosul, Iraq's second city in the north of the country.
Mother in Aid is part of the Chaldean Catholic Church. It's unclear what the church was being used for before Islamic State demolished it, because it appears on a list of churches presented to the US House of Representatives which were either "destroyed, occupied, converted to mosques, converted to ISIS headquarters or otherwise shuttered".
The Iraqi Journalists Syndicate (IJS) has also reported that Islamic State has murdered a female journalist who was working in Mosul.
Suha Ahmed Radi was working for a Mosul newspaper before the extremist organisation kidnapped her and then killed her on charges of spying. She is the 14th journalist Islamic State has killed in Mosul, according to IJS.
The organisation has said it does not know how long Suha Ahmed Radi was captured for, or how she was executed.
Islamic State captured Mosul in June 2014, forcing most of its Christians to flee.
Kurdish soldiers, militia groups and coalition airstrikes led by the United States have been fighting back against the extremist organisation in Iraq since it began capturing towns and cities in the country.