MongoDB - How to rename a field within an array of objects/sub-documents?

db.collection_name.find({

// Filter to only work on documents that contain the array field
"parent_field.child_old_field_name": { $exists : 1 }

}).forEach(
    function( doc ){

    // For each object in the current document's array
        for( i=0; i < parent_field.length; ++i ) {

            // Create the new field
            doc.parent_field[i].child_new_field_name = doc.parent_field[i].child_old_field_name;

            // Delete the old field
            delete doc.parent_field[i].child_old_field_name;
        }

        // Update the document
        db.collection_name.update(
        // Only the current document
        { _id : doc._id }
        // The updated document (as per the statements in the above "for" loop)
        , doc 
        );
    }
);

// This is how the collection that fits the example looks like:

{

    "_id" : "bs2XTNDYLSrp9Ae9m"

    , "parent_field" : [ 

        {

            "child_old_field_name" : "any value" 

        }

        , {

            "child_old_field_name" : "Any other value"

        }        

    ]

}