Give access to a subdirectory without giving access to parent directories

You are mistaken in your original assumption, which renders the rest of your question moot.

The minimum permission that a user would need on dir1 and dir2 is Traverse Directory. This will most likely be problematic to your users, though - so I would recommend Traverse Directory and List Folders. They will be able to navigate through the top two directories and get to dir3 where they have more permissions, but will not even see what files exist in the top two directories.

Permissions like Read & Execute and Modify are just collections of individual permissions. They're the first thing you see, because they're the most commonly used. If you need to get very granular (like this situation), click the Advanced button and dig into the options listed there.

Amazingly, if the individual has the full path to a subfolder on which they have at least R permissions, they require NO permissions on any of the parent folders, not even traverse. They can simply access it using the UNC. (They must, of course, have read permissions on the share; just not on any folders above the level they want to access).

I didn't believe this when I was told, but testing proves it out.

This is counter to what I thought I knew of permissions in the Windows world, and I suspect will be a surprise to many.

\server\folder1\folder2\folder3

If there are no permissions at all for Bilbo on folder1 and on folder2, but Bilbo has modify (for example) on folder3, \server\folder1\folder2\folder3 will take him right there, no problem.