How can I use Rules to 'Set a Data Value' on a node's field WITHOUT updating that node's 'updated' timestamp?
If it is not necessary for you to use rules, you can also try to invoke the hooks node_update and node_insert.
There you can load the parent, change the fields value and save it with the field_attach_update function.

You can find an interesting article about this at https://www.urbaninsight.com/2011/10/24/saving-nodes-fields-without-saving-node-itself