I can't imagine that Mustang doesn't have an interlock where the clutch must be fully depressed before the engine will start, but if it doesn't I can easily see how that kid would have done that. I'm guessing that Mustang has more than enough torque to start in first gear, and as soon as it started rolling, I'm sure the kid had no idea what to do and panicked.
I'm guessing the dad wasn't actually expecting whoever got in the car to start it and drive it, I'm sure he was just letting whoever it was sit in the driver's seat so they could feel all cool and whatever. He probably left it in first gear so it wouldn't roll, (I always leave cars with manuals in first when I park on flat surfaces, saves wear on your handbrake cable, and you don't have to worry about your handbrake cable freezing up in the winter, which does happen where I live.)
In short, both the father and the kid are at fault, the father should have been paying more attention to the kid in the car, like actually standing there and watching, and the kid is completely stupid to start the engine in the driveway not having any idea what he's doing.
Now, on to the handbrake comments. In many cars, the handbrake operates a small set of drum brakes behind the rear rotors, my car is this way, as are many others that I've inspected. On some, usually higher performance cars, the handbrake operates another set of calipers on the rear brake rotors, the Porsche Carrera GT and the Noble M12 GTO3R are like this.
On cars with drum type handbrakes, the handbrake may help a little in a last ditch situation with either faded brakes or a cut brakeline. A handbrake as always cable operated, so if your hydraulic brake system does get a puncture or lose pressure, the handbrake still works.
To whomever said that you don't need to use a low gear decending in the mountains has clearly never driven in the mountains, there is no way your brakes would survive otherwise. I have personally followed people in the mountans that don't know how to use a lower gear, and it is pretty frightening, seeing a truck's brakes on fire at the bottom of a mountain pass is enough to convince me....