Displaying a list web part from a parent site to a sub-site, or a sub-site to a parent site is a common requirement. SharePoint Online 2013 allows you to do this by publishing the list to the 'site gallery', at which point it is made available to add as a web part on parent or sub-sites within the same site collection.
The steps to do this are as follows:
1. Open the list in SharePoint Designer 2013.
2. Open one of the views in detailed view by clicking the view hyperlink.
3. Click into the detailed view between the "<WebPartPages:XsltListViewWebPart..." class as shown in the image below.
4. Click the 'Insert' tab.
5. Click 'Data View' and select the list you wish to publish from the drop down.
6. Click the 'Web Part' tab and choose Save Web Part 'To Site Gallery'.
6. When SharePoint Designer displays the following message:
"Do you want this Web Part to always show list data from the current web site…"
8. Go to the site where you need to display the list, edit the page, click the ‘Insert’ tab and then browse to the ‘Miscellaneous’ folder. You will find your list available to be added to the page.