I will add to what others have said.

Although your employer does not have to give you any protection it may be worth it to them to do so.

If you're looking at 2-4 weeks of leave, depending on your position, skills, the cost and time of recruiting, hiring and training a replacement may be higher than letting you go on leave, or at least unpaid leave. Not to mention a new hire is always a risk, they could be bad, they could quit in a year...

Promising that an existing employee has a position to return to after taking care of a sick family member creates loyalty and a good working environment, other employees see that

I would have a frank discussion about my needs and what the options are, tell them you aren't asking for a resolution during the conversation, you know they'll need time to explore options.

I also suggest you look at your position, could it be done part time and or remotely, will things fall apart while you're out or just get a bit behind. Could you offer to work a small number of remote hours while you're out if that would defray things?

Unfortunately there is simply a strong chance they could offer to protect your position but not be able to pay you. It's also possible they'd ask you to pay the insurance premium while on unpaid leave. You won't know until you speak to them.