Further details (and text article, and what appears to be an unedited dashcam video): http://kutnews.org/post/details-emerge-police-shooting-cisco-pet-dog?nopop=1
Okay so, first thing the cop does is order the man to put his hands up. What caused him to issue this order? Did Paxton appear to be a threat?
Then, in conflict with his first order, he tells the man "get your dog." However, he shoots the dog less than a second afterward, right after saying "dog."
Did he not see the dog at first? This would explain the conflicting orders. The dog must've been rushing toward the officer, otherwise there would've been more time between the order to restrain the mutt and the shooting.
Now put yourself into the officer's shoes: you're confronting a man you don't know, and a dog you don't know, and the dog is rushing toward you. The man, for all you know, just finished beating up his girlfriend. The dog, you know nothing about, it could even be a fighting dog. The dog is barking and running full tilt at you. You have half a second to react.
Even if the dog is friendly, if he jumps on you that's going to be distracting, may even cause you accidentally fire your weapon if you have it drawn (and watching the video further, he did already had it drawn), and would also provide an opening for the man to either escape or attack you. And if the dog isn't friendly, well...
It's tragic, to be sure, but I'm not really sure I can blame the officer.