Get parent category id from child category page for custom taxonomy

On a taxonomy archive page, be it for a child or parent term, you can get the current/queried term object/data using get_queried_object() which contains properties like term_id (the term ID) and slug (the term slug). And the property name for the ID of the parent term is parent. So you can do so to get the parent’s term ID:

$cat = get_queried_object();
$parent_cat_id = $cat->parent;

And for displaying a list of terms in the parent term, you can use wp_list_categories(). Here’s an example with the title_li set to '' and echo set to 0, which means I’m manually putting the output into an UL (<ul></ul>):

$cat = get_queried_object();

$list = wp_list_categories( [
    'taxonomy' => $cat->taxonomy,
    'child_of' => $cat->parent,
    'title_li' => '',
    'echo'     => 0,
] );

if ( $list ) {
    echo "<ul>$list</ul>";
}

If you want full control over the HTML, e.g. to add custom HTML before/after the term link or perhaps to add custom CSS classes, you can use get_terms() and loop through the term objects to display the term:

$cat = get_queried_object();

$cats = get_terms( [
    'taxonomy' => $cat->taxonomy,
    'child_of' => $cat->parent,
] );

if ( ! empty( $cats ) ) {
    echo '<ul>';
    foreach ( $cats as $cat ) {
        $url = esc_url( get_category_link( $cat ) );
        // change the 'before' and/or 'after' or whatever necessary
        echo "<li>before <a href="https://wordpress.stackexchange.com/questions/360088/$url">$cat->name</a> after</li>";
    }
    echo '</ul>';
}

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