It is not possible to view a whitelisted page through the MediaWiki API.
Steps to reproduce:
- Set $wgGroupPermissions['*']['read'] = false to create a private wiki.
- Add a page to the whitelist using $wgWhitelistRead = array( "Foo" );
- Access api.php?action=parse&page=Foo
- The API to return successfully, showing the contents of the page [[Foo]].
- The API returns a readapidenied error ("You need read permission to use this module"), even though a user can access index.php?title=Foo without an error.
This bug has been mentioned on the MediaWiki wiki since at least 2010 ([[mw:Special:Permalink/300785]]), but I couldn't find any existing filed bugs in the API category on Bugzilla.