Your textbook is probably using “went” as in left if it’s using partir. Using “aller” always requires a complement, such as “Je suis allé(e) au cinema avec mon ami.” (I went to the movie theater with my friend.) But you couldn’t say “Je suis allé(e). If you were just saying “I went with my friend” you couldn’t use aller. If you were wanting to say that you left with your friend you could use “Je suis parti(e) avec mon ami.” You could also use “s’en aller” which means the same. So if you wanted to say “I went with my friend” using aller you could “Je m’en suis allé(e).”

“Partir” is not always the translation for “to leave”. For example, to go on a vacation or on a trip is “Partir en vacances” and “Partir en voyage“.

It is implied that you leave your home to go on a temporary location, with the intention of coming back. In your example, I assume they went with their parents/friends on a trip or a weekend.

Both are correct, but the context may change, even if it’s minor.
At least it’s how I use it.

"Je suis allé chez mes parents" mean that you took the decision to go somewhere.

"Je suis parti chez mes parents" mean that you left a place before.