What your permit allows

Your permit normally covers only the job and employer named in it.

A new employer—or a new position with the same employer—requires a new permit before the new work begins.

A second job requires separate permission; the immigration office says sideline employment is limited and must relate to the main job. Read the immigration office work-permit guidance here: https://www.utlendingastovan.fo/en/permits/you-want-to-extend/your-work-permit