How to display the featured image of a parent on child pages?

You can use get_the_post_thumbnail() and pass the ID of the parent instead:

if ( $post->post_parent )
    $post_id = $post->post_parent;
else
    $post_id = $post->ID;

if ( $thumbnail = get_the_post_thumbnail( $post_id, 'post-thumbnail' ) ) {
    echo $thumbnail;
}