Was that PHP version installed using the Plesk autoinstaller or is it the result of an individual PHP installation that was added as an additional handler before PHP 5.6 became available through Plesk? I think it was the later, because in the default PHP package delivered by Plesk mysqlnd is included while this is not necessarily included when you compile PHP yourself or use an image that the provider or OS vendor supplied. It is possible, too, that on that system the database was upgraded outside Plesk and PHP drivers were adapted to the upgraded database or that you are using the OS vendor's PHP which accidentally happens to be PHP 5.6, too, but is missing some modules compared to the PHP that Plesk delivers. I don't know what exactly was done, but it has got to be something along this way.
Which way to choose to solve the issued depends on whether you want to manage PHP 5.6 version yourself or whether you want to use the Plesk delivered component. If you want to continue to use your current PHP you will need to recompile or add the mysqlnd module. How that is done depends on the operating system you are using. If you want to use the Plesk delivered PHP I suggest to deinstall the existing PHP 5.6 and use the autoinstaller to install the Plesk supplied PHP. That should include the proper mysqlnd module.