Obviously not, from the story: "the attorney general's office ... decided not to file charges." If there are no charges, there would be no arrest.
Just because they violated one or more TN laws by "providing" a loaded handgun to their child "recklessly", doesn't mean they MUST be prosecuted. Maybe they couldn't prosecute the parents without also declaring the child "delinquent" and subjecting him to an early introduction to jail.
If the child had been given proper safety training for what to do if the handgun is found, like a cigarette lighter or any number of other potential hazards, he would have been better off.
Duh! Of course they should be. Owning a gun comes with responsibilities. A five year old shouldn't even know where the gun is at. If you're going to have a gun around children there should be a trigger lock on the gun and it should either be in a safe or in a hiding place where the probability of a child finding it is unlikely.