Adding Child Terms Programatically – No Warning but No dice either

It doesn’t work because you use wp_insert_term function incorrectly. Please read the codex manual page carefully and you will see that the second argument is not parent slug, but your custom taxonomy.

So your function should look like this:

create_taxonomy_record($args) {
    $parent_term = 0;
    if (!empty($args[2]) && ($parent_term = term_exists($args[2], $args[1])) {
       $parent_term = $parent_term['term_id'];
    }

    $term = term_exists($args[0], $args[1]);
    if ($term == 0 || $term == null) {
        wp_insert_term(
            $args[0], // the term
            $args[1], // the taxonomy
            array(
                'parent'      => $parent_term,
                'description' => $args[3],
                'slug'        => $args[4],
            )
        );
    }

    else { echo "<h1>" . $args[0] . " exists in parent " . $args[1] ."</h1>";}
}

Pay attention to $args array, now it should has 5 elements:

1. Term
2. Taxonomy
3. Parent term slug
4. Description
5. Term slug

And you can use it like this:

create_taxonomy_record(array(
    'Label',
    'your_custom_taxonomy_name_has_to_be_here',
    0,
    'Label Related',
    'this_tax_term_slug',
));

/* ---> Create a child */

create_taxonomy_record(array(
    'Label 2',
    'your_custom_taxonomy_name_has_to_be_here',
    'this_tax_term_slug',
    'Label Related',
    'this_tax_term_slug_a_child'
));