Showing Children of page by name

You need to pur div container before while loop.
Here you go. Here is the modified code and container will have parent page slug class.

<?php

$args = array(
    'post_type'      => 'page',
    'posts_per_page' => -1,
    'depth'          =>  2,
    'sort_column'  => 'post_name',
    'post_parent'    => $post->ID,
    'order'          => 'ASC',
    'orderby'        => 'menu_order'
 );


$parent = new WP_Query( $args );

if ( $parent->have_posts() ) : ?>

  <div id="parent-<?php the_ID(); ?>" class="<?php echo $post->post_name; ?>-page">

    <?php while ( $parent->have_posts() ) : $parent->the_post(); ?>

        <h1><a href="https://wordpress.stackexchange.com/questions/203191/<?php the_permalink(); ?>" title="<?php the_title(); ?>"><?php the_title(); ?></a></h1>

    <?php endwhile; ?>

  </div>

<?php endif; wp_reset_query(); ?>