The next link is one hyperlink buried in the middle of the wall of text, and simply appends the new message to the existing one. It also differentiates between prev and parent?

It's super unclear.

Scroll down a bit farther to see the full comment tree.

"Next" goes approximately down the tree in the order it's displayed on the page, by depth-first search.

"Prev" just reverses the same process as "Next".

"Parent" differs from "prev" in that it goes to the parent e-mail even if this email has earlier siblings.

(Generally, I just scroll down to the tree view and click around manually.)

The page has four sections, divided by <hr> tags;

1) The email message, with a few headers included

2) A thread overview, with all emails in the thread

3) Instructions on how to reply

4) Information about how to access the list archives.

You need only care about (1) and (2). The difference between prev and parent is indicated by the tree view in (2). The previous one is the previous one in the tree, which might not necessarily be the parent if the parent has spawned earlier replies.