In Confluence REST API documentation it is stated that you need to use a Confluence Administrator user for adding a watcher to a page:

• Add a watcher to a page using REST API

  Create a new watcher for the given user and content id.

  User is optional. If not specified, currently logged-in user will be used. Otherwise, it can be specified by either user key or username. When a user is specified and is different from the logged-in user, the logged-in user needs to be a Confluence administrator.

Currently, if you use a space admin through REST API you are getting a permission error. Space admins can add watchers using Confluence UI. Therefore space admins should be able to add watchers to the page by using REST API.