I am not sure I understand the distinction between paralinguistic and extralinguistic.
Let's eat, grandma. Here, grandma is the adressee of the message, the actor (invited). Grandma is the one to eat.
Let's eat grandma. Now grandma is the direct object, the recepient of the action. Grandma is the one to be eaten.
With examples like prosody, pitch, volume, intonation I would also include emotion (when vocalized), style, pauses etc. These are paralinguistic.
Instead extralinguistic would be the fact that A and B are housemates therefore the phrase the "The doorbell is ringing" means "open the door".
Since written language does not seem to have prosody, pitch, volume, intonation etc and it appears to be a record of oral language these features would have to be marked with punctuation.
Punctuation seems to either be part of Syntax or Semantics/Pragmatics (but pragmatics deals with extralinguistic environment)
My question is "What is (the nature of) punctuation; where would it be studied (maybe syntax)?"