Subpages List Appearing on Every Page

If you want this for pages only, which would makes sense as to what you described this, just wrap the whole bunch of code in if ( is_page() ) { ... }.

If you were to put this into a function, you could just bail out early.

function wpdev_156446_list_parent_page_tree() {

    if ( ! is_page() ) {
        return;
    }

    // Your above code here
}

// EDIT

function wpdev_156446_list_parent_page_tree() {

    if ( ! is_page() ) {
        return;
    }

    $parent = get_post($post->post_parent);
    $parent_title = get_the_title($parent);
    $grandparent = $parent->post_parent;
    $grandparent_title = get_the_title($grandparent);
    $current_page_parent = ( $post->post_parent ? $post->post_parent : $post->ID );

    // is the homepage the granparent? = third level page
    if ($grandparent == is_page('0')) {
        $children = wp_list_pages("title_li=&include=".$post->post_parent."&echo=0"); // list the parent page
        $children .= wp_list_pages("title_li=&child_of=".$post->post_parent."&echo=0"); // append the list of children pages to the same $children variable
        if ($children) {
            ?>
            <ul class="submenu">
              <?php echo $children; /*print list of pages*/ ?>
            </ul>
        <?php
        }

    // is the homepage the parent? = second level page
    } elseif ($post->post_parent ==is_page('0')) {
        $children = wp_list_pages('title_li=&child_of=".$post->ID."&echo=0');
        if ($children) {
            ?>
            <ul>
              <li class="current_page_item"><?php echo get_the_title(); ?></li>
              <?php echo $children; ?>
            </ul>
        <?php
        } else {// your else stuff

        }
    }
}