With about a third of votes counted, the party was in third place with about 13% of the vote.
It would give the Pirates nine seats in the 63-seat parliament. It currently has three.
The centre-right Independence Party was in the lead with 30% and the Left-Green Movement second with 16%.
It is still too early to say which party or coalition of parties will be able to form the next government.
The early election was triggered by the resignation of Prime Minister Sigmundur Gunnlaugsson in April.
