Find Posts based on Child Post value

Child posts are related to their parents (through the “post_parent” field), and not vice versa. So you would have to query your child post type first, then get the parents of all the results.

// obviously these variable names and key names might not apply,
// change them as necessary
$performances = get_posts( array(
     'meta_key' => 'the_performance_date_custom_field',
     'meta_value' => $date,
     'post_type' => 'performance' ) );

// extract the parent post ID's from each of the returned performances   
$event_ids = array_map( 
     create_function('$post','return $post->post_parent'), 
     $performances );
$event_ids = array_unique( $event_ids ); // dump any duplicate events

// and finally, get the events from their ids
$events = get_posts( array(
    'post_type' => 'event',
    'post__in' => $event_ids ) );