wp_nav_menu: check if the list item has children and add a class to anchor link

I was able to add a “parent” CSS class to the anchor tag of menu items who have children by following this answer: Add ‘has_children’ class to parent li when modifying Walker_Nav_Menu

Here’s an example:

class Main_Nav extends Walker_Nav_Menu {
  /**
   * @see Walker::start_el()
   * @since 3.0.0
   *
   * @param string $output Passed by reference. Used to append additional content.
   * @param object $item Menu item data object.
   * @param int $depth Depth of menu item. Used for padding.
   * @param int $current_page Menu item ID.
   * @param object $args
   */
  function start_el(&$output, $item, $depth, $args) {
    global $wp_query;
    $indent = ( $depth ) ? str_repeat( "\t", $depth ) : '';

    $class_names = $value="";

    $classes = empty( $item->classes ) ? array() : (array) $item->classes;
    $classes[] = 'menu-item-' . $item->ID;

    $class_names = join( ' ', apply_filters( 'nav_menu_css_class', array_filter( $classes ), $item, $args ) );
    $class_names=" class="" . esc_attr( $class_names ) . '"';

    $id = apply_filters( 'nav_menu_item_id', 'menu-item-'. $item->ID, $item, $args );
    $id = strlen( $id ) ? ' id="' . esc_attr( $id ) . '"' : '';

    $output .= $indent . '<li' . $id . $value . $class_names .'>';

    $attributes  = ! empty( $item->attr_title ) ? ' title="'  . esc_attr( $item->attr_title ) .'"' : '';
    $attributes .= ! empty( $item->target )     ? ' target="' . esc_attr( $item->target     ) .'"' : '';
    $attributes .= ! empty( $item->xfn )        ? ' rel="'    . esc_attr( $item->xfn        ) .'"' : '';
    $attributes .= ! empty( $item->url )        ? ' href="'   . esc_attr( $item->url        ) .'"' : '';

    // Check our custom has_children property.
    if ( $args->has_children ) {
      $attributes .= ' class="menu parent"';
    }

    $item_output = $args->before;
    $item_output .= '<a'. $attributes .'>';
    $item_output .= $args->link_before . apply_filters( 'the_title', $item->title, $item->ID ) . $args->link_after;
    $item_output .= '</a>';
    $item_output .= $args->after;

    $output .= apply_filters( 'walker_nav_menu_start_el', $item_output, $item, $depth, $args );
  }

  function display_element( $element, &$children_elements, $max_depth, $depth=0, $args, &$output ) {
    $id_field = $this->db_fields['id'];
    if ( is_object( $args[0] ) ) {
      $args[0]->has_children = ! empty( $children_elements[$element->$id_field] );
    }
    return parent::display_element( $element, $children_elements, $max_depth, $depth, $args, $output );
  }

}