I'm dubious that this is a good translation of the English. It's more like

You don't have to tell that to your parents.

Yes you are right. 言わなくてもいいね is more like "don't have to tell".
However ～ってはいけないよ is common modal phrase, witch often express advice. In this case we can say also 両親にその事を言わないほうがいいよ。Literally translated means "You'd better not say that to your parents." but there is sense such as "You mustn't tell that to your parents."That sense depends on the situation.If we talk to unknown person sentence "その事を両親に言ってはいけないよ。"shall be a bit extraordinary and rude.
For that reason I propound two more translations.
両親にその事を言わなくてもいいね。 and 両親にその事を言わないほうがいいよ。
Both sentences sound like advise ,but in formal situation can be used to express imperativeness.
Japanese has a lot mutually replaceable expressions.

> For that reason I propound two more translations.
> 両親にその事を言わなくてもいいね。 and 両親にその事を言わないほうがいいよ。

But one of those _isn't_ a translation of "You mustn't tell that to your parents." The former also doesn't sound much like advice, because 'ね' sounds like the speaker assumes the hearer is already in agreement with him.

Yes again you are right. >ね' sounds like the speaker assumes the hearer is already in agreement with him.
However particle ね also makes imperative expressions softly and uncategorical , When we do not want obtrude o.'s opinions too much ,we use ね、よね、でもね、いいから,
Sorry for my bad English and awkward explanation.

So the English might be something like

You might want to keep that from your parents though, eh?
注意：カナダ語w

not

You mustn't tell that to your parents.

英文とのリンクを削除しました。
[#47258] You mustn't tell that to your parents.