With the following function you can return only the terms with a parent term (so are child terms), by post id and taxonomy. Notice that there terms also can be parents when using multiple levels of hierarchy.
/**
* Returns terms with parents of a post
*
* @param int $post_id the post
* @param mixed $taxonomy to get terms from specific taxonomy
*/
function get_terms_with_parent( $post_id, $taxonomy ) {
if ( empty( $post_id ) ) {
global $post;
$post_id = $post->ID;
}
$terms = wp_get_post_terms( $post_id, $taxonomy );
foreach ( $terms as $key => $term ) {
if ( $term->parent != 0 ) {
$new_terms[] = $term;
}
}
return $new_terms;
}