Why there are so few broken jaws is the same reason as in boxing: closed mouths. An open mouth more likely to suffer fracture, like happened to Ali against Ken Norton in '73.
Your dad suffered a much worse injury than he would likely ever suffer in a mixed martial arts competition. The back of the skull is a very sensitive area, and suffering a blow to it is very dangerous. Punching to the back of the head (a “rabbit punch,” as WonJonSoup said) is very illegal, as is spiking the opponent to the canvas on his head or neck. Fighter safety is taken very, very seriously. There does tend to be more blood than in boxing, mostly due to elbows, which are sharp. The cuts tend to be superficial ones that don’t require stoppage.
There are fewer punches in MMA for a number of reasons, the number of rounds being one. Range is another. MMA fighters tend to fight much, much more outside than boxers, because there’s more incentive to stay away from an opponent. Glove weight in the UFC is only 4 ounces instead of the 8 ounce or 10 ounce gloves typically used in boxing, so any one punch can do more damage. Not only that, but unlike boxing, the ref doesn’t come separate the fighters when they get in a clinch. If you’re in a bad spot in an MMA clinch, you’re subject to “dirty boxing,” clinch punches and elbows that would be illegal in boxing, as well as knees, which can be devastating, and takedowns, which can put you in the position of being submitted or under a fight-stopping hail of strikes. This makes fighters very, very careful about getting into clinching range.
Ironically, all of this probably makes MMA safer than pro boxing. In boxing, a downed fighter is given a ten count to rise to his feet and is then checked by the ref to make sure he’s capable of continuing. A stunned but not downed fighter may be given a standing eight count by the ref to see if he’s capable of continuing. In MMA, a stunned and downed fighter is pounced on, usually resulting in stoppage right then and there. Not only that, a an unstunned but downed fighter can be submitted, ending a fight when the fighter is actually uninjured. This means longer fights for the boxers, more blows, and more possibilities of concussive injury.
I don’t know if it’s bourne out by medical science, but what you’re taught in the gym is that it’s not the one clean knockout punch that’s dangerous, it’s the blows that come after that punch. If you “get your bell rung,” training for you is over for the day. If a MMA fighter gets his bell rung, the fight’s most likely over for him. If a boxer gets his bell rung, he might keep fighting.
The conception that MMA is mostly entertainment like pro wrestling is unfortunate. Mostly, I think, it’s because of how it’s being marketed. It’s presented as spectacle rather than an athletic contest, to the degree that people are lumping it in with pro wrestling. Despite that, every single bit of the hype and spectacle dissappears when the door suts and the bell rings.