Add a 'parent' CSS class to WordPress wp_nav_menus

add_parent_class_to_wordpress_menu.php

<?php
/* Add a function to add a parent class to WordPress parent menus */
function sennza_add_menu_parent_class( $items ) {
	$parents = array();
	foreach ( $items as $item ) {
		if ( $item->menu_item_parent && $item->menu_item_parent > 0 ) {
			$parents[] = $item->menu_item_parent;
		}
	}

	foreach ( $items as $item ) {
		if ( in_array( $item->ID, $parents ) ) {
			$item->classes[] = 'parent';
		}
	}

	return $items;
}
add_filter( 'wp_nav_menu_objects', 'sennza_add_menu_parent_class' );