Multiple archives to display parent and subcategories content

With the below, you are grabbing terms from the event_category taxonomy
Note that you are always grabbing the same terms though.

$terms = get_terms( 
    'event_category', 
    array(
        'orderby'    => 'count',
        'hide_empty' => true
    )
);

If you want to grab only from the current pages children, then you should grab the current term and set your term list to only grab children of the current term:

$current_term = $wp_query->queried_object;
$terms = get_terms(
    'event_category',
    array(
        'orderby' => 'count',
        'hide_empty' => true,
        'parent' => $current_term->parent
    )
);

Here you are cycling through them…

foreach( $terms as $term ) {

    // Check and see if the term is a top-level parent. If so, don't display it.
    $parent = $term->parent;
    if ( $parent!='0' ) {

        // Define the query

What are you trying to do with this query?
You are creating it, but then doing nothing with it.

        $args = array(
            'post_type' => 'events',
            'event_category' => $term->slug
        );
        $query = new WP_Query( $args );

This just writes out a link:

        echo'<a href="'.home_url().'/big-events/'.$term->slug.'"><span class="cat-name">' . $term->name . '</a>';

It might be better to do something like:

        echo '<a href="' . get_permalink($term) . '"><span class="cat-name">' . $term->name . '</span></a>';
    }

}           
?>

rewritten a bit:

<?php
$current_term = $wp_query->queried_object;
$children = get_terms(
    'event_category',
    array(
        'orderby' => 'count',
        'hide_empty' => true,
        'parent' => $current_term->parent
    )
);
if ( !is_wp_error( $children ) && is_array( $children ) && 0 < count( $children ) ) {
    foreach ( $children as $child_term ) {
        $child_link = get_permalink( $child_term );
        echo <<<HTML
<a href="https://wordpress.stackexchange.com/questions/176032/{$child_link}"><span class="cat-name">{$child_term->name}</span></a>
HTML;

    }
} else {
    # No child terms were found
}
?>

Related Posts: