Update: Holy crab, thank you for all your fantastic input!
Thanks to your suggestions, I'll be going with Syncthing; it pretty much fulfills everything we need.
I'm also Nextcloud-curious, but will try that some other time (for this issue it feels more like using a sledgehammer to crack a nut). Will tweak Syncthing here and there to suit my parents' needs, and then we're good to go.
Thank you so much, everyone, I super appreciate you taking the time to help me! 🙏
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