I was not concerned about the status of the father, but rather that there are two distinct thoughts embedded in the second sentence. (1) My father was watching TV. (2) My mother was sitting beside my father.
Yes, OK, you say the thoughts are distinct, and I say the 2nd thought is not being used to define the subject of the first. Those views are similar enough. Nevertheless, we have established punctuation rules which tell us that the commas should not be omitted in such a case. Therefore there is no need to appeal to arguments based on "aid to comprehension" or nuances of pausing.
...punctuation has to be inserted as an aid to comprehension. In my view, punctuation gives grammar its human face, by indicating how we breathe and pause infinitesimally when we articulate a sentence.
I don't like the sound of this. To my mind, what distinguishes us from the apes is the ability to communicate our thoughts not only through uttered sounds but through the written word, which has the advantage that it can transcend generations. In a sense therefore I find it more appealing to think of the spoken form as being a rendering or interpretation of the written form, not the other way round. I consider our written language, not our spoken language, as our primary language. So our spoken nuances should reflect the punctuation, not vice versa. I accept that this view is a bit unorthodox.
Our armoury of established punctuation rules should normally suffice to tell us where to place and omit commas. If, having exhausted those rules, our written meaning is still insufficiently clear without needing to tinker with the punctuation on the basis of what we think our written text should
sound like when read out, then re-phrasing is probably a better option.