If you can't have another chicken, I'd really suggest you try to find a new home for her where she can be with her kind. The mirror idea
@Tiana's chickens suggests can be useful for separating a hen whilst she's injured or sick, but long-term, chickens need socialisation from other chickens. Chickens kept alone can feel very skittish due to the fact that, naturally, they look out for each other, warning each other of danger and subtly vocalising their feelings; think of all the noises they make when they lay an egg, when they see food coming, when they've found a nice treat, when they see an interloper or possible threat? Alone, your hen has to do all of that watching out by herself, which can get stressful for her.
Maybe you can sweet-talk your parents by telling them how lonely she must be without a flock to take care of her?