How to display selected taxonomies by their parent

It’s not so hard, but some PHP coding will be needed this time. Here’s the code that will do the trick (assuming that there are only 2 levels in that hierarchy):

$post_terms = get_the_terms( $post->ID, 'services' );
$terms = array();

foreach ( $post_terms as $term ) {
    if ( $term->parent ) {
        if ( ! array_key_exists( $term->parent, $terms ) ) {
            $terms[ $term->parent ] = array(
                'term' => get_term( $term->parent, 'services' ),
                'children' => array()
            );
        }
        $terms[ $term->parent ]['children'][] = $term;
    } else {
        if ( ! array_key_exists( $term->term_id, $terms ) ) {
            $terms[ $term->term_id ] = array(
                'term' => $term,
                'children' => array()
            );
        }
    }
}

foreach ( $terms as $term_info ) {
    echo $term_info['term']->name;
    if ( ! empty($term_info['children']) ) {
        echo ': ';
        foreach ( $term_info['children'] as $i => $child ) {
            if ( $i ) echo ', ';
            echo $child->name;
        }
    }
    echo "<br/>\n";
}