At what age do we start to buy Insurance for hospitalization or death? Is 24 years old too early (given that parents are the only dependents who are financially independent)?

Insurance for hospitalization should always be bought as early as possible. Children are not immune to hospitalization.

Death coverage however, is a nice to have if one does not have liabilities.

If one's parent's are dependent on a working child providing income streams, then the child should insure himself/herself. If not, it's not strictly necessary.

Hi. Rather than age, I think we should be looking at financial capability. Insurance should be bought once we start working and are financiallly capable.
Do note that premiums at younger age only applies for death insurance, so if you bought it at 20 yrs old, you would paying less than someone who purchased it at 40 yrs old, assuming equal coverage and riders, same gender, etc.
For hospitalization insurance, premiums increase every year regardless of when the insurance was bought, so even if you purchased it 20 years old, when you are at age 40, you would be paying the same as someone who first purchased the insurance at age 40.

But at age 40, our health condition may or may not allow us meet the criteria for purchase.

In short, it's good to start young but only when we are financially capable. Just my two cents' worth.