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

Here is Why You Need An Adapter

The outlet and plug type used in New Zealand is I and in Barbados are A and B. This is how they look like.

Barbados

In Barbados, outlets of type A and B are in use.

Type A

Type B

New Zealand

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

Type I