"Parsing" means, in this case, processing text (originally it had to with linguistics and syntax/grammatical analysis, but as with many other words, the electronic age has expanded its usage). Having that preference means that if I just type a URL like this:
http://flyertalk.com/forum/forumdisplay.php?f=457
it will automatically be turned into a clickable URL. If it weren't, you'd have to select the text, copy and paste it into your address bar. What a pain!

This way I don't need to bother with the "URL" forum code or the link button on the message composing toolbar, though I still might if I wanted to link arbitrary text like this (though usually I would want the linked text to specify what I'm linking to. do as I say, not as I do! ) I'm not sure why one would not this preference enabled, as it is by default. The old board always behaved this way, as have all other modern ones I've seen out there in use.

vCard is a standard for exchanging personal/contact information. If I sent you my full vCard from my computer's address book, it could include physical addresses, home, work and cell numbers, e-mail addresses, web addresses, company name/title/department, a photo, birthday/anniversary dates, relative/friend names, and on it goes. In this particular case, however, all it does is give your member name and e-mail address. This does expose your e-mail address, so for privacy reasons it is not enabled by default. It is, however, the only way for people to e-mail you, it seems, with the new board, as it says "Sorry! That user has specified that they do not wish to receive emails.", even if you've specified the preference to allow mail from other users, as I have. It seems, however, that private messaging is really a good way to communicate without needing any personal info at all. The days of "YGM" may be ending - or being replaced by "YGPM", perhaps?

"MyFlyerTalk" on any page, or http://flyertalk.com/forum/usercp if you want to bookmark it directly, will take you directly to the list of forums to which you've subscribed. This will also show new subscribed threads - not all, but rather only with new posts since you last read them - as well as newly received "reputation"/feedback comments about your posts, as long as the feedback system continues to generally exist for the time being.