What I'm trying to do seems rather simple, but I can't find a way to do it.
Imagine somebody sends you a link for a dropbox folder. You can go to that URL and see all the files in the folder.
I'm trying to write a script in either python, php, or javascript to get all the download links in that folder from that URL.
How would I do that? Without logging in/authenticating into the account of the user who sent the link in the first place.
Assuming that the contents of the folder are indeed public as you've indicated in the comments, you might want to look into the use of PHP's file_get_contents() function.
From the docs:
file_get_contents— Reads entire file into a string