Since version 11 Parallels Plesk Panel has its own MultiPHP solution for selecting different PHP versions on a per-domain basis. You can read about it here. It allows end-users to select not only the PHP version but also a PHP Handler that will be used for PHP processing.
This makes integration of CloudLinux PHP Selector in Plesk more difficult than in other panels. The PHP Handler selected in MultiPHP by the end-user must be compatible with PHP Selector requirements, otherwise, PHP Selector will not work. Plesk MultiPHP solution will overwrite the version in case the incompatible PHP Handler is selected there.
Plesk MultiPHP solution can be found under a user's domain -> PHP Settings.
So, go to that menu and make sure that only compatible PHP Handlers are used, among those are FastCGI, CGI or LSPHP served by Apache. If you see that PHP-FPM is selected in field, change it to one of the supported ones.
This is an example of a correctly set up PHP Handler:
Press the Apply button at the bottom of the page.
If you are a server admin and you want to allow your users to select only CloudLinux PHP Selector compatible PHP Handlers, you can do it. Just log in as Plesk Administrator -> go to Tools & Settings -> PHP Settings and disable PHP Handlers you wish to disallow for your users. If you don't want to confuse your customers with a lot of different PHP-related menus and you want them all to use PHP Selector, you can also disable Plesk's MultiPHP solution altogether. It can be done as Plesk Administrator -> Service Plans -> [Service plan name] -> Permissions tab -> Pshould be disabled.
Now that we have set up the correct PHP Handler, we need to make sure that PHP Selector doesn't have a native version set. The thing is that if you select native in PHP Selector, no matter what version is native for you, PHP Selector will stop working. As a result, the PHP version selected in Plesk's MultiPHP (domain -> PHP Settings) will be used instead.
So, this version selection is wrong:
Selecting any other version will overwrite the Plesk's MultiPHP settings. This selection is correct:
You can disable the native version as any other version completely from PHP Selector in LVE Manager >> Selector tab.
That's it, the PHP version of your websites should be fetched from CloudLinux PHP Selector now.