Is it possible to do multiple internal monologues in parallel?

No one can speak two words or can sing two songs in parallel.

PHYSIOLOGICALLY

There's something called polyphonic singing, which overtine singing is a form of. Here's a demonstration of overtone singing and a demonstration that also includes undertone singing, so it's possible both physiologically and cognitively.

For verbal communication, some languages require just little enough use of parts of the mouth and/or throat such that it's possible to produce sounds in parralel that can build up words.

Overtones and undertones are not the only techniques that can be used to produce muliple sounds at the same time that are enough to communicate in a verbal langauge. For example, beatboxers produce multiple sounds at the same time. It depends on the language. For example, some language requires a certain pressure in the lungs to produce some of it sounds such that other sounds cannot be produced at the same time, while in others that is not the case, while some require so much that no sounds can be produced in parralel that are a part of verbal languages, , as far as I know, such as whistling languages (and possibly clicking languages but I'm not fully certain about that).

There's also non-verbal communication that can be done. In this case, single-handed sign language can be done with 2 hands at the same time. This might limit communication, but, again, it depends on the language.

In addition, other physical communication can be used for conversing.

COGNITIVELY

I do not know enough about neuroscience to answer this, except for what I heard in documentaries from some researchers (and I don't have citations nor links for that) that the human brain can retain only about 3-7 "small" pieces of information at the same time (I've heard an estimation in bytes, too, however I don't know how relevant and/or accurate it is, as meaning can be encoded to very large and very small amount of bytes), and I do not know enough about how language processing works in the brain to answer you that, and self-conversation also requires processing of the own self, so it's not only the language processing part(s) of the brain that will need to be involved, as there will be needed to be some feedback loop that takes own-produced information as stimuli for processing - something that is usually done serially, and not in parallel.

So I do not think that we have enough cognitive capabilities for that, but like I said, I'm not an expert in the field.