You Do Need power plug adapter while traveling from Palau to Cook Islands.

Here is Why You Need An Adapter

The outlet and plug type used in Cook Islands is I and in Palau are A and B. This is how they look like.

Palau

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

Type A

Type B

Cook Islands

In Cook Islands, outlets of type I are in use.

Type I