I assume that uploaded files are rather small and internet is fast, otherwise tasks of checking urls alive and parsing pdf documents should be handled by standalone workers processing some queues and don't deffer response.
The requested form with ability to upload files is just at the bottom of documents list page then accessing service via web browser.
https://stackoverflow.com/questions/31436357/how-to-extract-all-links-from-pdf-file
https://www.metachris.com/pdfx/
https://github.com/metachris/pdfx
https://github.com/heroku/python-getting-started
https://github.com/encode/rest-framework-tutorial
https://github.com/alanjds/drf-nested-routers