You Do Need power plug adapter while traveling from Faroe Islands to New Zealand.

Here is Why You Need An Adapter

The outlet and plug type used in New Zealand is I and in Faroe Islands are C, E, F and K. This is how they look like.

Faroe Islands

In Faroe Islands, outlets of type C, E, F and K are in use.

Type C

Type E

Type F

Type K

New Zealand

In New Zealand, outlets of type I are in use.

Type I