With the article un/una both are considered correct as mentioned in the lesson. It is with el/la that only the masculine form is admitted: el ave, un ave, una ave.
I agree- what's the reason?
The reason to use el and not la in front of these words is to make it easier for their pronunciation, so there aren't two "a" in a row "la agua" - it's easier to say "el agua".
As for the reason to admit both un and una, there is no known explanation and the RAE doesn't expand on it. Although it's a lot more common to say "un ave", the Academy doesn't penalise "una ave" (but it is less common).
Here's an explanation of what they say.
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