Looking to set my parents up with Channeks DVR and figured Apple TV would be the simplest option for them. There's been quite a few models over the years and should like to do this as cheaply as possible.
Are there any benefits to Channels DVR by using a more recent model?
The ATV4 is the minimum you need for the App to run. It works for Channels just fine.
ATV 4K has much better ethernet throughput, if you are hooking it up that way.
To add to what @djcastaldo said, if you're going to use ethernet for the Apple TV, the Apple TV 4K's ethernet is much faster. This leads to way better seek times.
Because the original Apple TV 4th generation uses a 100mbit ethernet, it leads to slower seek times than if it was on WiFi. That's only if your wifi is faster than 100mbit of course.
The problem with the Apple TV 4th gen is that it puts you into the predicament of getting better seek times by being on wifi, which most people in the community will agree, is less desirable than ethernet. Using wifi means that it sucks up a lot of bandwidth while watching TV, leaving not as much available for other devices. It's reliability can also be effected by other devices on the wifi. And mostly, people just don't trust wifi over a wire 
Thanks all. Sounds like we need the 4k.
Everything @maddox says is correct. But I use a regular ATV4, and it works perfectly with Channels, even with its slow ethernet connection 
So depending on your needs and the amount you want to spend, either may work for you.
I have 8 ATV's and 4 of them are ATV 4K. As I got new tv's with 4k I upgraded to 4k. As said here already they have a faster Ethernet Port but also a faster CPU. If I were you I would just get the 4K. The 32k version ($20 less) is also fine. Not sure what the older 4 cost. Keep in mind if your parents like it they may get more of them so maybe move the 4 to another room if that is what you end of getting.
I have a mix of both of the two most recent Apple TVs (Apple TV 4 and Apple TV 4k) in my home. All my Apple TVs are hard wired and the Channels app works very well with both of these models with the only appreciable difference being skipping forward or backward. That said, for the price difference of $30 retail I suggest the more expensive 4k version. In addition to faster ethernet, the 4k has a 2 generation newer CPU (A10X vs A8). The newer CPU is much faster and a difference maker for certain applications.