Display a sub site list on a parent site in Sharepoint Online 2010 and 2013

If you want to show a list belonging to one site on another site (in the same site collection), you can use a list template. BUT if you want the list to dynamically reflect changes made in either site on both sites, here’s how:

Insert the sub site list on a sub site page
In Sharepoint Designer, open the page and select the list
From the ribbon “save to gallery”
Choose the option to show items from THIS list (not a list relative to the site the web part is inserted on).

Go to parent site and insert that web part on a page.
This works for an Enterprise account.

In Sharepoint Online 2013, this still works, but Microsoft in their own inimitable way has made it a bit harder and less obvious.
Using Sharepoint Designer 2013, it’s not apparent how to “save (a web part) to gallery”. The trick is to open a page with the list view on it in SPD, click anywhere in the list view web part, right click and select “tag properties”. Click “OK” and the web part tools will show in the ribbon.
From there you click “save to gallery” and the option about showing “list data from the current web site or from the same list relative to the site it’s inserted on” will appear. Opt for “list data from the current web site”, save the web part and insert it (from the “miscellaneous” group) on the other site…