I have recently undertaken a personal project to translate my WIP novel into Japanese in parallel with writing it. I am managing both in LaTeX and editing them in vim, but the topic itself is too specific for either crosslisting, I feel. Is there any good software for writing translations of works that are themselves changing over time in parallel? (I would normally cp the file and translate paragraph-by-paragraph, but I would lose any progress I made so far of the translation, or be translating out-of-date source material due to the edits I am working on in the original). Besides that, any advice?
No, there is no software that would automate the translation of newly written paragraphs for you
This is not something you would normally do. You might think that you are "going in one straight pass" as you wrote in the comments, but in the end you will find something that needs to be edited. Some things that don't add up. Some things that don't sound right after being away from the first chapter for a few weeks/months. This is normal.
But the problem is that every edit of yours will more or less invalidate the translation you already have, meaning that you have to do that again. And what happens if you edit the first three chapters, translate each one after you edited it and with the fourth chapter you realize that there is a bigger change necessary to the first two? Now you have to edit the first two chapters again - which means you have to edit the translation yet again.
That's why the normal process is to write a novel, edit it and then start with the translation once everything else is done and ready to be published. Or better yet, once it is already published. There will probably not be any major differences once your novel is out in the world, so it's pretty safe to assume that you won't have to do the translation two/ three/ four/ ... times because of something that changes in the original work.
Without any accurate software to automatically translate your work into Japanese this is not a useful approach. And if you are thinking about recommendations for translation software - well, look at google translator. Would you really be satisfied with those results? You would very likely have to rewrite the entire automatic translation. It might help a bit, but with software you could easily just put it all in after you are done with your work.
If you feel like you want to present the Japanese version to people speaking Japanese because you feel like they would be able to give you better feedback on what to do better than you might want to start asking yourself: would it be better to first write the novel in japanese and then translate it into english/...?
Doing the translation in parallel is likely not a good way to work on your novel and there does not seem to be any software that would reliably help you with that style of working on your novel. I personally recommend against translating your work-in-progress novel.
Since you are using commandline tools already, I will assume that you are familiar with various Unix tools and able to use them.
Keep your documents in git or another revision control system. Then you can use the diff functionality (commandline or any of the various visual tools around) to see which parts are new or changed.
If you number your paragraphs, from there it is easy to identify the part in the translation that needs to be updated.
You could also move your workflow to a writing tool, like Scrivener or Ulysses or any other one. There you can work chapter-by-chapter and have a number of options on how to mark which chapters are new or changed.
I work by myself a lot with plain text (or markdown), and prefer it for quick notes and such, but for serious writing, I use a writing tool, exactly for such functionalities (marking chapters with tags, reordering easily, etc.)
You didn't give a specific reason for why you work in vim, that's why I'm including this as a suggestion.