There are various types of leave. Earned leave aside, many companies make provisions for "Leave Without Pay" (LWP). When an emergency situation arises and the employee does not have any leave balance, he/she can apply for LWP. Therefore, you may talk to your HR and ask for LWP on compassionate grounds. If you are working in the same city where HR is located, then go and talk to them personally. If HR is located at a different station, then talk to them on the phone and explain your problem. Hopefully, HR will show understanding. If HR also shows no understanding, then you will have no option other than resigning from the job.
As of now, you are attending to your filial duties. Problems of your kind are common among lower/higher middle-class individuals. Nevertheless, we have to strike a balance between filial duties and career requirements.
From the company's side, they are unable to grant any more leave because most companies operate with lean manpower. There is no provision for standby manpower as in government departments. Hence, their inability to extend your leave.
Both parties have their own perceptions. However, when perceptions differ, situations of this kind arise.
All the best!
Regards, Dinesh Divekar