Should you list even brief employment on a job application?

I am a bit OCD and scrupulous about these things.
I just applied for a job and saw that the background check might include previous employment history, and I am not sure what that will entail. I have a completely clean criminal record, and as far as education is concerned, I will be graduating with honors this year.
The issue I have is that previously, I haven't had much steady employment. Both of my parents have been extremely ill throughout the years; my mother has battled cancer twice and my father has had many medical issues that have put him in the hospital as well. As a result, I have often spent most of my time at home caring for them - and my employment history has suffered. I spent nine years working for a family as a babysitter/nanny, so that made it to my resume; I also worked in customer service for almost a year in a family business that eventually closed due to the health issues I stated above. I am embarrassed to put that I've worked four weeks here, 8 weeks there, etc. So I've omitted them from my resume (not thinking that it would be much of an issue at that time). Now I am scared that this long, unsteady history will show up if and when they investigate me further. It's embarrassing.
I've googled employment checks and some sources state that they can only do a check on the jobs you have listed; others say that everything will show up. So I am not sure what to believe now. Can someone give me some insight into this? I'd greatly appreciate it. Thanks!
Should you list even brief employment on a job application?
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