your menu is pages oriented.
pages have parents and the query asks for the parent and the siblings.
When accessing posts, posts have no hirarcy, wp_list_pages will not work and you cannot query posts by parent.
The big question is what you’d like to display when accessing a single post…
if you want to display the main site navigation yuo can do this:
<div id="dep-menu">
<?php global $post;
if (!is_front_page()) { // so I don't display a link on the homepage saying "Homepage"
if (is_page()) {
if($post->post_parent) { //if page has parent
$children = wp_list_pages("title_li=&include=".$post->post_parent."&echo=0"); //include a link to the parent
$children.= wp_list_pages("title_li=&child_of=".$post->post_parent."&echo=0"); //add the child pages
} else {
$children = wp_list_pages("title_li=&include=".$post->ID."&echo=0");//parent link
$children.= wp_list_pages("title_li=&child_of=".$post->ID."&echo=0");//children
}
} else {
$children = wp_list_pages("title_li=&echo=0&depth=1"); //list top level pages
}
if ($children) { ?>
<ul>
<?php echo $children; ?>
</ul>
<?php }
}
?>
</div>