I have a KML of local authorities in England and Wales. The research I've done suggested that Bing Maps does not support KML.
What are my options for overlaying / drawing these boundaries onto the Map? I've tried this here to try and parse the KML, but that didn't work for me either.
You are correct in that the Bing Maps V8 JavaScript control does not currently support KML. However, this is planned to be added in the first half of 2017.
If you want to add KML to Bing Maps V8 right now, there are a couple of options. The first is to take a look at the open source KML module for Bing Maps V8 here: https://github.com/koldyr/BingKmlParser
The second option is that the Bing Maps Spatial Data Services supports KML files. You can upload a KML file to this service and it will expose the data as a spatial REST service that you can easily pull into Bing Maps V8. Note that this is a data service and doesn't store any of the style information from the KML file. So this is only a good option if you are only concerned with accessing the data in the KML file.