I ended up using this code:
<?php
global $post;
$children = get_pages( array( 'child_of' => $post->ID ) );
if ( is_page() && $post->post_parent ) : ?>
This is a child-page.
<?php elseif ( is_page() && count( $children ) > 0 ) : ?>
This is a parent-page (with one or more children)
<?php else : ?>
This is a parent page without children.
<?php endif; ?>