No free parking onsite but parking on the street should be no problem.
Parking in public spaces is free during night and on Sundays. The hours that public parking is charged are as follows: Mon-Fri 9 am to 6 pm, Sat 10 am to 4 pm, Sun free. Please note that you are requiered to buy a ticket at City of Reykjavik vending machines - credit cards accepted. There are three different zones, and it is practical to know that P1 is the most expensive zone (our street), and P2 is half the price (the street below). P3 you find down the hill by the sea, practical if you are leaving the car there for the whole day.
Still another option is the moderately priced parking house opposite our building, open 7 am to midnight.