WordPress display only child category in template

If you know the parent category name, then you could just do an if inside the foreach

Demo code

<?php
foreach((get_the_category()) as $category) {
 if($category->cat_name!="parent category name")    { 
  echo '<a class="tag-cat ' . $category->slug . '" href="' . get_category_link($category->cat_ID) . '">' . $category->cat_name . '</a>';
 } 
} ?>

?>