Do I have any rights in refusing what my employer's requesting?

That's tricky but I am thinking you have no choice. Except...you work every weekend and that's a terrific bargaining chip. It's one more day a week (unless they break up the 8 hrs into two days), they did cut your hours on your request so they have cooperated in the past, so maybe it's your turn to compromise.
They may not want to give up someone who covers every weekend however.
But as employers they need what they need to run their business; often their needs come first.

My sister worked 20 hours a week so she could be home with her daughter and had a hissy when her work wanted her to go up to 25 hours. (she had been supposed to go back full time when her daughter was one year old, but put it off for three years). She was surprised, I don't know why, when her job told her either 25 hours or you find another job. She decided to quit. Unfortunately, your work looks at what they need, not what you need in your personal life.

So to meet your employer's needs, you'd have to hire in care to cover you at home two days a week, is that right?

Who covers you over the weekends? Is there any possibility that that person could help?

In UK law - bear with me, I know it won't be the same - an employer is required to hear employees' requests and to meet their needs for flexible working as far as the business can accommodate them. But they're not *obliged* to, and that's in the UK/EU, and I would be astonished if US law was more family-friendly. So although it's always worth checking what your state says about this kind of thing, I'd be very surprised if you found out that the law was on your side and that your employer can't insist you re-increase your hours.

First of all, add up. Figure out what it is going to cost a) to provide additional care for your mother and b) for the business to let you go and hire a half-time employee to replace both you and the other person. See what sense you can make of the numbers, and then you'll be ready to negotiate with your employer. For example, if you scratch their back by agreeing to the increase, what (apart from the extra money earned) can they offer you? Or, if you've looked at it from all sides and you just can't see a way, will it make financial sense for them to let you go? Cash isn't the only factor: they won't want to lose an experienced, committed and loyal employee unless they have to, either.

And, as Geevesnc points out, do look ahead: is this a long-term change or a stop-gap until they have a chance to hire someone else?

Best of luck, and I hope you can find some way of making it work. Don't give up your job lightly.

Yes, you can refuse and then they can decide to let you go. Perhaps you can hire someone to come in to take care of mom a couple of days a week? Check into county resources for elderly. Maybe, after a few weeks they will let you go back to 17 hours.