If you are able to sideload the app and get a parsing error then either the apk is corrupted or not compatable with your devices.
Have you tried downloading the apk from another site, or sideloading an app such as aptoide and see if you can get a working version of the TV Control app from there?
Also if you are happy with sideloading apps why not perform a google search and sideload the googleplay store onto the Fire tablet, there are plenty of walkthroughs and videos available. Doing that should give you the full access to all the apps currently available. Keep the Amazon store as a back up for their regular freebies.
The Service you do for others is the rent you pay for your room here on Earth - Muhammad Ali