If money is not a problem I would recommend Apple TV.
Apple TV has the advantage of begin wireless, and as you said, your parents will be able to control it from their seat. This quite useful for watching watching shows, but even better for other things, like looking at pictures on your iPad on the big screen, for example.
On the intuitiveness, if your parents already know how to access the multitasking bar at the bottom of the iPad it is pretty easy: just a switch away. Practically speaking, easier than dealing with walking, plugging it in, and having to stand up to input things to the iPad, or dealing with a long cord.
Apple TV has many other advantages/features: Photo Stream through iCloud, Netflix, Youtube, flickr, Internet Radio, WSJ, Movie Rentals, TV Show Store, etc. I don't know how many would your parents use/need, and of course Movies and Shows come for an extra cost, but its something to keep a thought on.
It's $40 dollars more expensive though. If your parents don't need such features, they don't mind a cord, and if they don't really need controlling their iPad while displaying something on the big screen, than it's some good $40 to save, but if you can spare it I think you get plenty of extra features and wireless commodity from it.
Best Answer
As other commenters have already said, it doesn't seem to be possible to AirPlay a display to two targets at once.
Option 1)
The best solution I can think of would be to run something like AirServer on a Mac or PC. Attach the computer to both projectors and have it mirror the display. AirServer will make the computer a valid AirPlay target and you'll be able to mirror your iPad's display to it (and thus see it on both projectors simultaneously).
Option 2)
You could also buy an HDMI splitter and two long HDMI cables. That way you could AirPlay your iPad's screen to an Apple TV and have the Apple TV output to both projectors simultaneously.
Option 3)
I believe it's also possible to connect some projectors to a network (eg: via Ethernet or Wi-Fi), so perhaps you could look into that? You may be able to create an AirPlay endpoint on your network (perhaps with a tool like the aforementioned AirServer) and then direct its output to your network-connected projectors.
Option 4)
You could browse the App Store for an app which streams camera input from an iOS device to the network. You could essentially turn the iPad into a streaming HTTP server and then have any number of devices tune into it. Then you could attach an iPad, iPhone, Apple TV, computer (or anything, really) to each projector and have them stream from the source iPad (eg: By visiting the URL for the source iPad's stream in a web browser like Safari or a video player like VLC). This has the advantage of not requiring any long cables, although you'd need a solid Wi-Fi connection.