"No, dice la donna."
What does this sentence mean? No native speaker would say, "No, the woman says." as a complete sentence.
Here are various interpretations I can think of, together with what I'd expect the Italian to be:
"'No' says the woman" / "La donna dice: 'no'" "No, the woman speaks" / "No, la donna parla" "No, he says the woman [is the one]" / "No, dice che è la donna"
Is the meaning less ambiguous in Italian?
Not really; your second example would indeed be "parla" and not "dice", but the first and last are valid: "No, says the woman" or "No, he's speaking of the woman".
Duolingo accepted my translation of "No, says the woman" for this sentence.
Personally I think the reason we find this sentence odd is not because there is anything wrong with this sentence, but simply because we are so used to seeing this kind of sentence in the past tense, that it looks odd when written in the present tense. This makes it an issue of pragmatics not syntax.
Think about it... in written English you come across sentences like "No, said the woman" and "No, the woman said" a lot, but you would hardly ever come across the present tense of those sentences ie "No, says the woman" or "No, the woman says". But that's because in English when we describe what someone says we almost always use the past tense, so using the present tense sounds weird.
Of course you could possibly argue that Duolingo shouldn't teach sentences that are not in common usage, but any language course is a series of building blocks that are build on top of each other... Teaching this sentence here leads on to teaching the past tense sentence "No, the woman said" in later lessons.
It's generally "to say" or "to tell", i.e. "lui dice che sta bene" (he says he's fine), "mi ha detto che sta bene" (he told me he's fine), but it can occasionally mean "to speak" in some idiomatic constructs such as "dice la verità" (he speaks the truth).