I can't download Pokemon Go?
Your phone running Android 4.3 makes it incompatible with Pokémon Go. Your best bet is getting a new phone, since the S4 isn't getting newer updates, or using a custom ROM.
If you want to use the APK, you will need to allow unknown sources. This is what causes the "parse error" when you try to install it. To allow unknown sources, do the following: Go to Settings > Security. Scroll down until you see "Unknown sources" and switch it on. Press OK when it gives you a warning. Now reopen the APK and it should install fine.