How to show page children links ONLY when viewing parent

WP doesn’t do it out of the box… but
https://www.ads-software.com/search/show+children+pages?forums=1

Yes.

In your Page template file (page.php), get the ID of the parent page (this code must be in the loop):

<?php $page_id = get_the_ID(); ?>

Then, wherever you want to output the list of links to child pages:

<?php
$page_children = get_page_children($page_id, '');
if (is_array($page_children)) {
    echo '<ul>';
    foreach ($page_children as $child) {
        echo '<li><a href="' . $child->guid . '">' . $child->post_title . '</a></li>';
    }
    echo '</ul>';
}
?>

Slight correction to last post!

You don’t need to get_the_ID in the loop, and you should check if $page_children is empty, not an array. So, with that said:

<?php
$page_children = get_page_children(get_the_ID(), '');
if (!empty($page_children)) {
    echo '<ul>';
    foreach ($page_children as $child) {
        echo '<li><a href=\"' . $child->guid . '\">' . $child->post_title . '</a></li>';
    }
    echo '</ul>';
}
?>