Is my advisor being unreasonable by not approving classes in a different field than his?
It all boils down to your advisor’s estimate of your capacity to complete your PhD in time. Perhaps your advisor sees your newfound love for NLP as a distraction from what should be your main focus: successfully defending your thesis within the span of X years. If you are well on the road to graduation and are exploring a side project then the restriction may be unfounded, though not unjustified (perhaps further establishing yourself in your field would improve your career prospects). 

Of course, no one can stop you from independent study or taking a Coursera class, but your advisor’s reluctance likely stems from the concerns I mentioned above. This should also be something you shouldn’t take lightly: there’s often good reason to listen to one’s advisor.

Good luck!