Your PHP installation appears to be missing the MySQL extension which is required by WordPress.

Symptoms

Accessing your domain returns the following:

 Your PHP installation appears to be missing the MySQL extension which is required by WordPress.

Searching for the PHP extension MySQL, we see that this is already loaded.

[ root@server ~]# php -m | grep mysql
mysql
mysqli
mysqlnd
pdo_mysql

Description 

The error will occur when the configuration value "suPHP_ConfigPath" is found within the user .htacess. 

To identify this we recommend searching for the variable "suPHP_ConfigPath" within the domains document root.

[ root@server ~]# grep suPHP_ConfigPath /home/$user/public_html/.htaccess
suPHP_ConfigPath /home/$user/public_html/php.ini

Note: Please replace "$user" with the actual user that is facing the issue.

If the above search returns a suPHP_ConfigPath configuration value, then this would most likely be the reason for the original error.
The reason for this is that the suPHP_ConfigPath configuration value overrides the default php.ini's. It will ensure to only use the PHP configuration configured in the php.ini specified within the suPHP_ConfigPath.

Workaround

To resolve this issue, you'll need to comment the suPHP_ConfigPath entries from the .htaccess file in question. Doing so will load the correct php.ini's and allow the proper PHP extension(s) to be loaded.