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

Here is Why You Need An Adapter

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

Réunion

In Réunion, 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