Yes, it is ambiguous.
This is a very common situation in Latin:
Two different semantic roles are expressed with the same grammatical case — or two morphologically indistinct cases.
Here the person to suffer from the verb nocere is in the dative, and so is the person with the obligation to not nocere.
The same can happen with many different cases.
Distinguishing "the father's photo of the son" from "the son's photo of the father" is hard, as both the owner and the object of the photo would be in the genitive.
In an accusativus cum infinitivo structure both the subject and the object are in the accusative.
It is in my opinion no stretch grammatically that the sentence is ambiguous.
Nor is it, in my opinion, a stretch semantically.
Often the context restricts the world of possible readings to a very small selection, but here two paths seem to remain open and there is a difference in meaning.
But either way, the moral seems to be that nobody should harm anybody else, so even "no one should harm anyone" would work fine.
Whether or not the translation matches the original in grammar is irrelevant; what matters is content of the message, and that is thankfully unambiguous enough.
The only strong reason I could see for favoring one reading over the other is comparison to other similar maxims within those fables.
If they tend to be structured in a specific fashion, then we may want to supply e.g. tibi or otherwise understand who is under obligation.