Resume Accomplishment, but Highly Sensitive

I would answer this with the lawyer's favorite answer, "It depends..."

Are you a recent grad looking for your first job?  If you don't have much else to put on besides your undergrad degree and summer jobs, this could be good.  You could discuss managing a budget and your parents appointments and healthcare decisions.

Have you already established a career?  I wouldn't put it on if you have other, professional related accomplishments to discuss.  It will take space from the things that will more directly show what you are capable of in the workplace.

In either situation, you need to be aware of what the employment laws are in your country as well as biases.  While discussing your personal situations should not be a reason to disqualify you from a position, some managers may not want to hire a person who will need to take time to manage thier parent's health.  Not hiring someone for this reason is illegal in many places, but it would be difficult to prove.

It mostly comes down to what you are comfrotable revealing in the first few interactions with your potential new employer.  If this is a major part of your life and will be something that needs consideration, you probably should be up front in discussion but not on papers.  If it's something that just happens and doesn't affect your work product, it may not be worth bringing up until it will actually impact you.