Learn Spanish (Spain): Mis padres ya no se hablan; discuten mucho. - My parents no longer speak to each other; they argue a lot.

Here, se acts as a reciprocal pronoun, which is used when two or more people do an action to each other. Because the verb hablar is in the plural form hablan, adding se makes it mean they speak to each other, rather than they speak to themselves.

How can I make it extra clear that they aren't speaking to themselves?

Context usually makes reciprocal actions obvious. But if you want to be absolutely clear that the action is mutual, you can add the phrase el uno al otro after the verb to explicitly mean to each other.

What does ya no add to the sentence that a simple no would not?

The phrase ya no means no longer or not anymore. It shows that a change has happened. If you just said no se hablan, it would mean they do not speak to each other in general, but ya no se hablan emphasizes that they used to speak before but stopped.

Doesn't discuten just mean they discuss something?

The verb discutir is a very common false friend for English speakers. While it can mean to discuss, it is much more frequently used in Spain to mean to argue, fight verbally, or bicker. To say discuss in a calm, friendly way, you can use hablar de or charlar sobre.