I started my PhD in CS last fall in a lab doing one particular field (neurological-based deep learning) since that's what I was interested in when I was in undergrad applying and for about the first year of grad school. Since then, I stumbled into NLP and I find it fascinating I've been reading a lot of papers and I think I'd like to pursue it for my academic career – I've even started attending lab meetings with another lab in my department doing NLP. I'd really like to take a formal course in NLP, especially one taught by the PI of this other lab, but my advisor needs to approve any course I take, and he won't sign off on any course that isn't in his field. He says there's no point in me taking a course that won't benefit my work in his lab. Is he being reasonable here, or should I be able to take courses in tangential fields?
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.