In a statement issued on Thursday, the DPK accused Kosumi of paying 30,000 euros for the charter of a plane aboard of which he returned from Turkey where he spent his holidays.
Kosumi, who took up office four months ago after the departure of former PM Ramush Haradinaj for The Hague, claims that the charter was not paid with government funds but by his friends.
The DPK has urged UN Civil Administrator Soren Jessen-Petersen to investigate the case which it says harms the reputation of Kosovo and questions the stability, democracy and responsibility of local institutions.
The request was supported by another opposition party, the ORA, led by Veton Surroi.
Kosovo government spokesman and media advisor Daut Dauti on Thursday night resigned from his post explaining that he was forced to make that decision because of the difficult situation in the prime minister's office.
The scandal was given wide coverage in most Kosovo Albanian-language dailies on Friday. The province's government and the UN Mission have not commented on the accusations against Kosumi yet.