I took a look around and found this thread on apple’s discussion site:
Apple recently added a setting to the Apple TV that allows for only user-verified devices to connect to the Apple TV. Trying to connect with AirParrot when this setting is turned on will result in this message being displayed: This AirPlay connection requires iOS 7.1 or later on iPhone or iPad, or OS X 10.9 or later with the latest iTunes on a Mac.
Can you try these instructions from that thread:
To disable the setting, go to Settings > AirPlay > Security > Require Device Verification and turn it off. This setting was introduced in Apple TV software version 6.1. More on device verification can be found at the Apple support portal.
We are in the same boat as software like AirParrot–talking to AirPlay while not being an official Apple product. I bet your AppleTV has this setting turned on.
Please let us know how it goes. If you’re successful, we will make sure to publish this workaround more clearly and make our support people aware.