You can create an internal url system that redirects to the partner sites. Depending on what you need to do, you could create a node (content) with info about the partner site where you stored the link to it. Then instead of displaying the link directly on the site you could instead display the url:
redirect/[nid]. The url would be known by your module, and you could store info that you were sending some one over to the partner site before loading the node to find the url to send them over to.
You would have to create a custom module for this, but the above method could be a way of doing this.
You could also use tools like Google Analytics to track outgoing links instead. It depends where you want the data and what you need to do with it.