This should work to get the parent’s tags.
<?php
global $wp_query;
$attachment_id = $wp_query->post->ID;
$parent_id = get_post_field('post_parent', $attachment_id);
$parent_tags = wp_get_post_tags($parent_id);
$tag_count = count($parent_tags); // Counting the tags to find know the last one so there is no pipe
$i = 1; // Setting up the count
if ($parent_tags) { ?>
<div>
Tagged with:<br />
<?php foreach ($parent_tags as $tag) {
if($i < $tag_count) {
echo '<a href="https://wordpress.stackexchange.com/questions/173304/. get_tag_link($tag->term_id) .">' . $tag->name . '</a> | ';
} else {
echo '<a href="https://wordpress.stackexchange.com/questions/173304/. get_tag_link($tag->term_id) .">' . $tag->name . '</a>';
}
$i++;
} ?>
</div>
} ?>