I need to transform one array into another.

The first array describes a message. The message has its own ID and the ID of its parent.

$firstArray[id]["parent_id"] = parent_id

I need to transform this into an array which describes which parent messages have which 
children:

$secondArray(parent_id) = array(id, id, id, id);

So, for example:

$firstArray[4]["parent_id"] = 2;
$firstArray[6]["parent_id"] = 2;
$firstArray[12]["parent_id"] = 6;
$firstArray[13]["parent_id"] = 12;

into

$secondArray[12] = array(13);
$secondArray[6] = array(12);
$secondArray[2] = array(4, 6);