LINK is for when you want the exact location of a comment. For instance, if I'm talking on meme and saying "wait, I'm not that nonnie. I'm the one, who said this: http://spnanonhaven.livejournal.com/12197.html?thread=55756709#t55756709", I'd right-click on LINK and copy the link location. (This comment wasn't me, btw.)
You click on PARENT when you want to read the comment that the comment you're currently looking at is replying to. (In the example above it would be this one: http://spnanonhaven.livejournal.com/12197.html?thread=55755685#t55755685)
You click on THREAD when you just want to see the single thread or subthread and not all the comments on a page.
Re: Help with LJ Page links - dumb questions
You click on PARENT when you want to read the comment that the comment you're currently looking at is replying to. (In the example above it would be this one: http://spnanonhaven.livejournal.com/12197.html?thread=55755685#t55755685)
You click on THREAD when you just want to see the single thread or subthread and not all the comments on a page.