I am trying to change the order of text elements through arcpy, such as sending them to the front or to the back. I know I can change the position of text elements using arcpy, but I can't find any references to changing a map elements order in relation to other map elements?
My understanding is that ArcPy does not have access to the order in which layout elements draw/display in ArcMap.
Consequently, I think that this is something that you would need to pre-author into the map using ArcMap, so that the order is already what you need in ArcPy.
You could submit an ArcGIS Idea to have this enhanced but, if you do, I suggest targeting ArcGIS Pro rather than ArcMap. ArcPy with ArcGIS Pro does not seem to have access to the draw/display order of layout elements either.
I suspect that the way to describe such an enhancement would be to ask for a layout property which was a list of element objects in draw/display order.
As PolyGeo said ArcPy does not have access to the order but if thats needed you can clone text elements in desired order so in the end you can get what you seek. Not good but its working way but i don't know if thats possible for your situation.