A parking card for people with reduced mobility can be applied for through Ísland.is with electronic ID.
For the application, both for the first application and renewal you need:

• A medical certificate for a parking card sent electronically by a doctor.

• A passport picture of the applicant. A picture from the driving license registry can be used or a picture can be uploaded.

If applying for a child, parents submit the application with their ID and select the child in question. A photo of the child must be uploaded in the application.

There is no fee for issuing a parking card.

Further information and applications for parking cards can be found here; parking card for people with reduced mobility.