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Sub-tasks are not items that go into a sprint. Of course, they do really, but only because they are part of their parent issue, which can be in a sprint.
The only way to get your sub-task into a sprint is to re-open the parent and bring that into the sprint. Or move the sub-task so that it is part of a different issue that is going into the sprint.
I'd also take a look at your process, to prevent you landing in this logically nonsensical place. You have broken data when you say "issue done, but sub-task not" - the sub-task is part of the parent, so how can the parent be done when part of it is not?
That would be nonsense, as a sub-task is part of its parent.
Technically, sub-tasks don't even go into sprints, we just treat them as such because they are a part of their parent.
But, yes, you can split them - by moving them to different parent issues so they can be part of things that do go into different sprints.
It's nonsense in Scrum. You don't take bits of a story between sprints.
It might well work for your team, but that does mean you're not doing Scrum and will need to recognise that a Scrum tool such as Jira is for Scrum and so is unlikely to be that helpful for non-Scrum process.
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