Many comparative politics works lump the Icelandic system in with the “parliamentary” democracies.* Today we are reminded of why that is incorrect.
As part of the fallout from the “Panama Papers”, the Prime Minister, Sigmundur Davíð Gunnlaugsson, requested that the president, Ólafur Ragnar Grímsson, dissolve parliament and call early elections. The president refused the request.
Now, one could argue that this act alone does not prove the system is not “parliamentary”, because some presidents in strictly parliamentary systems might be able to refuse a dissolution request under some circumstances. But the event does show that the Icelandic president does take politically consequential actions, and on top of the direct election of the post, this surely qualifies the system as “semi-presidential” in practice as well as in formal rules. That is, rather than being purely ceremonial and accepting “advice” from the head of government, a president who is an agent of the electorate has exercised discretion.
The prime minister subsequently resigned.
* That is, those works that don’t just ignore it.