How to make code compatible with gevent?

Question

Python

Hey! Happy upcoming Holidays to all, especially those who have this holiday, mostly Labor :)
I have some misunderstandings with the transition to gevent. I am writing in Django and everything seems to be more or less clear there: I connected all the necessary modules to work with gevent, pro-mock-patched everything as it should be, made a connection pool with the base that supports gevent. Everything works, parallelized.
Now directly in the business logic of the application. What should I do? I have views that take a long time to parse some documents in response to a request. For two or three seconds. Sometimes even 10 seconds. Processing uses third-party libraries.
If I understand the essence of gevent correctly, then parallelism works exactly until a code is encountered that completely blocks the execution of greenlets. Those. if any external library encounters long calculations, then my application will stop accepting external connections until the gevent-incompatible code ends, right? Or will this greenlet simply be blocked with a call to this library, but other connections will still be accepted?
If my first guess is correct, then the next question is: what is the way out of this impasse? Rewrite third party libraries to support gevent?