Errr.... HTML parsing is done by browser. You can use fgets() to get first X lines of HTML source.
You might try to find some library that would (on the server with the list) process HTML and send you pure text. Maybe you can try combining it with lynx.
Or, if you just want links, make a script that would parse the document with eregi(). Like this:
S - your server
L - server with links
On L: parse.php - parse out all the links you want (with eregi() or lynx or some 3rd party libarary functions). As this is a high traffic site, you can save this data to temporary file. Next time, compare timestamps of file with links and temporary file. If temp is newer, fine - use it's data. If not, re-parse the file with links.
On S: call parse.php from L.
I hope I understood properly what your problem was and that this is of some use... :-)