Certainly from the outside it is indistinguishable, but that's not the conscious experience of it. I can type the words in the buffer without thinking about them at all. It is possible that at that point they are no longer language, depending on how my brain actually does this. However, the conscious experience of it is that "I" need to decide what to say in both conversations and I can only experience that decision a conversation at a time, even if the thoughts themselves are generated and/or in parallel.

A similar effect occurs when playing music. I can only consciously work on improving a single thing at a time. What does happen over time is that what a "single thing" is can become more encompassing. For instance, with the piano I had to first concentrate on what the chord I wanted was, then concentrate on hitting each note of the chord. Now I can just play that chord, without thinking consciously about the notes involved or how to hit them. So if a song calls for a C chord, I can just focus on "hit a C chord" rather than "hit the notes C, E, and G". Then the C chord itself grows into its various inversions, major/minor, arpeggiation, etc.

But at no point in my entire life have I ever had the conscious experience of multiple conscious experiences in parallel.