Gonna go out on a limb here and say this groom is NOT the a**hole.
If I'm being totally honest, I would be livid if my parents decided not to attend my wedding because my sister's dog was sick. First of all, that's no reason for the parents not to attend. If the sister can't make it because of the emergency then so be it. Also YOU HAVE OPTIONS. I know a dog being sick is very serious, but so is your son's wedding. Board the dog, take the dog to the vet, ask a friend to watch. Take the dog with you! There are airbnb's for this kind of thing.
All of that being said, I wouldn't be talking to my parents for the foreseeable future. There's no excuse.
Oh and the video that the best man created, roasting
them online? That's gonna stay up for a while. Cry about it all you want, but at least he showed up on the most important day of my life.
Verdict: Not the a**hole