The party button toggles part mode, which is some kind of auto-playlist XBMC features. Very handy in my opinion, use this a lot myself.
The party mode button on the iPhone was dropped in favor of the edit button, which IMO is more important to have.
With the latest version (finally) featuring editable menu items on the remote control, you can add your own party button:
Use the generic http command, and enter:
...replace a.b.c.d with your XBMCs IP and 8080 with the correct port.
This will use XBMCs old http api, which will be dropped in one of the next XBMC versions - but should work for now.
Direct integration of one app into another (or parts / plugins whatever) isn't possible on iOS.
I've thought about a "marriage" with airvideo before. Even posted in their dev forum about a year ago about an url scheme - which would allow me to deep link into the airvideo app and thus switch to airvideo and have it jump right into the movies directory - but the guys at airvideo won't add this due to security implications (there are indeed some)
@all you guys out there with sorting issues:
the next release will respect the sorttitle, that should fix it.