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

Here is Why You Need An Adapter

The outlet and plug type used in Saint Barthélemy are C and E and in Cook Islands is I. This is how they look like.

Cook Islands

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

Type I

Saint Barthélemy

In Saint Barthélemy, outlets of type C and E are in use.

Type C

Type E