How to sync a taxonomy between Drupal and a 3rd-party system? I have a custom made system (non Drupal) that has a taxonomy like:

Element 1
   Element 1.1
   Element 1.2
   Element 1.3
Element 2
   Element 2.1
   Element 2.2
Element 3
   ...


And I have a Drupal site that I would like to be able to:


Use this taxonomy for tagging content, and 
Add more terms that the non-Drupal system can use too.


Is there a popular way to sync a taxonomy in a Drupal system with an external system?
You can import taxonomies, and even nested taxonomies, with CSV data generated from your source data. I have been done before and it works quite well with module Feeds.

As for your taxonomy system, you must generate dynamical CSV files (in fact, a PHP or a Drupal node with some custom code), which feeds the Imports you generate in Drupal and which are associated to a vocabulary. In that way you can generate taxonomies with children.