No, the narration in Japanese does not specify who is saying these words.
If we take a look to the sentence (I tried to provide a more literal translation):
そんなおとぎ話をよく聞かされたっけ That kind of fairy tale was told quite frequently
We see that the verb 聞かされた is used. This is the short form of the past, causative-passive conjugation in Japanese of the verb 聞く (to listen). In causative-passive constructions, the agent is marked by the particle に. For example:
寝る時、母に話しを聞かされる。 [I am] told a story by mom before going to sleep.
Since in your original sentence there is no agent marked by に, there is no way to know who is the one making the warning from the text itself. In Japanese, it's pretty normal that some information is understood directly from context, and therefore passive or causative-passive construction without the agent specified are common.
It you are interested in the causative-passive form, please check out this. When 聞く ("to listen") is conjugated to the causative 聞かせる, it becomes "to make someone listen", i.e. "to tell someone (a story, etc). The same logic applies to the verbs 見る "to look" and 見せる "to make someone look", i.e. "to show".
Then, if you turn the (short) causative 聞かす into the causative-passive 聞かされる, it literally means "someone is made to listen" or in a less convoluted way "someone is told (a story, etc)".
All in all, I'd say that "they told us" is not a bad translation, but I'm not an English native speaker. Maybe other expressions such as "They say..." or "It is said that..." are better. Also note that the English translation is somewhat liberal, for example it throws in "forevermore" even though nothing like that is said in the Japanese version.