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

Here is Why You Need An Adapter

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

Saint Barthélemy

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

Type C

Type E

Cook Islands

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

Type I