It depends on which exercise brought us here. There are several exercices on the same grammatical point and they all leads here. So, if someone gets the exercise of writing down in french what he/she hears, then both sigular and plural forms are possible since they sound exactly the same.
I understand that the present tense in French can be used for to mean "She speaks" and "she is speaking." Is there another, more specific tense for "She is speaking" (present continuous tense, I think its called)?
That is a good question. The answer is that the French does not have a verbal form equivalent to the English continuous tense (be verb-ing). But you have an expression which means exactly the same idea of an action in progress: "être en train de".
So, "she is speaking" = "elle est en train de parler".
I just don't understand how french people can tell in spoken language if it's singular or plural with er verbs!
When someone says "elle parle/elles parlent", it means that pronouns "elle/elles" replace a noun or a name. So you will know who the speaker is talking about the same way as you do when you hear "she/they".