I definitely would NOT put the dog to sleep.

Still, keeping in him a home where there is a child too young to understand respect for him and the parents can't be bothered to train this child otherwise is not really safe. You can understand him not wanting a toddler laying on him or landing on him etc...many dogs would snap at children for this, heck, they would snap at their own pups for this. To dogs, this is perfectly normal behaviour, their version of discipline. It is nothing to do with the dog being vicious, it has everything to do with the dog simply being a dog.

I doubt he intends any harm on the children, the scratches they recieved would have been accidental. Pups have fur covering their faces but human children do not but a dog can't tell the difference.

If they want to keep the dog with them, their only choice is to drill it into the younger child not to lay on him or jump on him etc...
They should never ever leave a young child unattended with a dog even for a second anyway and they MUST always be supervised so that they are there to stop the toddler if she tries to do something they think will upset the dog.
Even toddlers of 2 can learn simple things like this. In fact, you'd be surprised at what they can learn. People get the wrong idea that children as young as this are too young to understand anything so they are too soft on them and let them get away with things or even laugh at bad behaviour like when they smack an adult but the child is learning a lot, they are learning how soft their parents are for example so a few years down the line, they have an uncontrollable little brat that throws tantrums etc....the price to pay for being too soft on a toddler you don't think understands anything. At least when they are a toddler, you don't need to use as much force to get your message across like you'd need with an older child who got too used to getting away with murder. If you get it into their heads early that they cannot push you, they will be much better behaved when they are a bit older.

No child should be left unattended with any dog for any amount of time under any circumstances anyway and they should stop her if they see her go to do anything that could end up in tears.

Also, they mention that the child was laying on his back and he growled but she didn't get up so he snapped at her. Why the hell did they leave her laying on him if he was growling. Didn't they figure it was time to get the girl off of him before he lost his temper?