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)?

It is never too early because we can get sick or die at any time.

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.

By age of 24 as long as you have an income, and in good health - get reasonable insurance coverage that match your needs. Please make sure you can sustainably pay it all on top of your living expenses, financial commitments, etc. Reason being, you'd have more options to change or choose policies, while being deemed as low risk to the insurance companies.

Have you consider when... it be too late? Apart from death (choi!), and as we age - it is possible some of us are struck by medical conditions that would become permanent - rendering us with a pre-existing condition (e.g. Diabetes, etc). On such situations, your insurance options be greatly limited as you are a, higher risk individual now.

So, periodical review of policies and having sufficient insurance coverage while you are healthy with income, is only prudent.