I'm the founder of Kimmelot German Shepherds, and it sounds to me like you are not a strong enough leader for your dog. So, what she needs is another owner.
German Shepherds were bred as Sheep Herd dogs. Meaning that they were bred to keep preditors away from sheep, like wolves, foxes, coyotes, stray dogs. The wolf was bred into them not only to make them devoted and to give them the desire to follow an alpha leaders directions, but also to be territorial. It is their natural tendancy to not allow a preditor in their territory, which is as far as they can see. It requires lots of control by an alpha leader, and lots of early dog socialization to curb and control this instinct.
Herding drive , is really prey drive controlled. When a herding dog like a Shepherd or Collie is rounding up sheep or other animals, they are really hunting them, with a leader. They just don't move in for the kill, but allow the alpha leader to choose when that is done, out of respect for the alpha leader, and you don't have that respect.
Same as a wolf leader signals with their eye or a nod of their head for this wolf to go left and that wolf to go right, they follow the leaders directions to bunch up the animals, or to help pick one off. Herding dogs have to be taught, strongly, to curb their biting. So again, going after things like chickens is natural prey drive, that just hasn't been curbed by a strong enough leader.
I don't think your parents want the liability of a strong natured, uncontrolled , high prey drive dog around , that does have a taste for blood, has learned to kill, and does not take your directions. That is fair.
But before you put her down, talk to an expert like me, call some German Shepherd breeders, or rescues, or guard dog services, and see if you can't get her placed into a more appropriate home. Guard dogs in a service is not a bad home, during the day they live in a regular kennel, always with a partner dog, at night they are driven to say a car dealer or someone with a wearhouse they don't want broken into, and the spend their nights, patrolling or just being a deterrant to crime. Guard Dog services usually charge $ 700. a month for a pair of dogs. This would be a better life than being put down.
You also say she has not been aggressive to people, so maybe she'd be ideal for a single man, who doesn't live with other animals, as a companion. Or a single woman who feels like she needs some protection.
You can contact me at firstname.lastname@example.org we do rescues of German Shepherds and others and might be able to help you place her.
35 years raising , 7 generations of Kimmelot German Shepherds, training, showing, bording, rescuing.