I've never used the AirDisplay App or AirParrot and AirPlay recently stopped working for me, too.
Platform Info:
* I have _no_ third party apps for display mirroring. Never tried them.
* I have a brand new iMac27. It's only two weeks old. It was working fine and stopped within the last day or two.
* I have late 2011 Macboo Pro that has worked for about a month. It won't work now.
* I have tried an iPhone 4S and it doesn't work.
* I'm using a wired Ethernet connection to both my iMac and AppleTV.
* I'm using Wifi on my MBP.
* Both iMac27 and MBP are on 10.9.2
Behavior Info:
* Sometimes AirPlay works, but usually it won't.
* All platforms seem to fail in the same manner: try to connect, timeout, and give up. AirPlay "availability" follows the AppleTV AirPlay "enable/disable" setting quickly and accurately, but ultimately, no connection can be made.
* I have tried rebooting my iMac and Apple TV multiple times. Both on their own and together. It doesn't help.
* When I select Airplay from the menu bar, the icon lights up, nothing seems to happen for a while, then I eventually get a "Could not connect to Apple TV" dialog.
* Communication between the two boxes seems solid in some ways. The AirPlay icon goes comes and goes quickly and accurately as I enable/disable Airplay on my Apple TV.
Summary:
I'm kinda wondering if an Apple TV update is to blame? My Apple TV is on automatic upgrades. It is only a month or so old and I only use it for AirPlay.