Can I access node ID of the parent node of a field collection field in field-collection-item--FIELD_NAME.tpl.php
Load the field collection using entity_load by passing the entity id. It gives you object where you can get the Host entity of the loaded entity. example profile2 will have user entity as host entity, or commerce line item will have commerce order as host entity. 

$entity = entity_load('field_collection_item', $your_file_collection_id);
$host_entity = $entity-&gt;hostEntity();
echo $host_entity-&gt;nid;