Is there a way to auto-upload photos to Flickr on Mac?
For an OS-independent solution, the following IFTTT recipe looks like it will do what you need.

Upload to dropbox - push through to flickr

In a nutshell, the recipe monitors a specific folder in Dropbox (the recipe uses "/Flickr", but you can change that) and uses the "Upload public photo from URL" action on the Flickr channel to upload the image.

There are a number of recipes that use Dropbox and Flickr, so even if this one isn't exact (and you don't want to customize) you should be able to find one you want. Here's another possiblity: Move Dropbox photos from Camera Uploads folder to flickr gallery.