Dropdown Category form. Only show Sub-Categories

You just need to set the parent argument of get_categories() (or better yet, get_terms() to the ID of the the-building-code term. If you don’t know the ID you’ll need to retrieve that first:

<form class="js-filter-form">
    <select name="categories" id="categories">
        <?php
        $parent = get_term_by( 'slug', 'the-building-code', 'podcast_category' );
        $cat_args = array(
            'taxonomy' => 'podcast_category',
            'parent' => $parent->term_id,
        );

        $categories = get_categories($cat_args); ?>
        <option value="">All</option>
        <?php
        foreach($categories as $cat) : ?>
            <option class="js-filter-item" value="<?= $cat->term_id; ?>">
                <?= $cat->name; ?>
            </option>
    <?php endforeach; ?>
    </select>
</form>

Also note that there’s no post_type argument for get_categories() or get_terms(), so I removed it.