WordPress still has mysql_query() in its codebase (as of February 27, 2018).
mysql_*() was deprecated in PHP 5.5.0 and removed in PHP 7.0.0. Since WordPress still supports PHP 5.2 (for reasons beyond understanding), it needs to support both mysql_*() and mysqli_*(), WordPress will use mysqli_query() when available ( PHP > 5.3.0 ) and fallback to mysql_query().
If it’s the WordPress $wpdb object that is using mysql_query() it means you’re running on PHP 5.2.x, which you really should upgrade. If you’re certain you’re on PHP 5.3 or greater, then there’s some other code that’s using the old mysql_query().
Although based on the error, it looks like it’s a problem with the permissions in your SQL database.