Show all sub categories?

To exclude parent categories from your term list, you can just pass over them after you have retrieved your term list with get_terms

Example:

<?php
$args = array(
  'orderby' => 'name',
  );
$terms = get_terms( 'mytaxname', $args );
 if ( !empty( $terms ) && !is_wp_error( $terms ) ){
    foreach ( $terms as $term ) {

        if( 0 == $term->parent )
            continue;

        echo '<a href="' . get_category_link( $term ) . '">' . $term->name . '</a><br/>';
    } 
}
?>