I concur that you should list it and not leave the gap in time. That gap looks bad. You might even be able to provide some keywords in a short description besides the job title "Caring for an Elderly Parent". Some skills that an employer may find useful such as scheduling, financial planning, estate preparation, medical consulting, or similar. Don't call it babysitting, it is patient care for example.
I would also cover the orig 5 month gap as the whole 11 months as parent care. It may not have been full time, but I am sure you were doing some of these during that time while you were actively looking for work.
I used to have a handle on life....... but it broke!
Semi-Retired 7/1/16: working part-time (60%) for now