All perfect praise be to Allaah, the Lord of the worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.
Sick leave, to the best of our knowledge, needs a medical report to prove that a certain employee is sick. It is meant to alleviate difficulties from the employee himself, not from his children. By no means would you get it if your children are sick, except through lies and forgery, which are undoubtedly unlawful.
What you should do, if your children are sick, or any of your relatives or neighbors who need somebody to serve them when they are sick, is to explain the circumstances to your work authority and give them the complete picture, without distortion. They would, almost certainly, respond in accordance with the current status, in a way that allows you to do your duty towards the sick person on the one hand, while avoiding any critical situation at work or lying on the other hand. If they do not respond, then make a request for a normal or even necessary leave. If that is not available, and the patient is in dire need for you, and your need to work is necessary or almost necessary, then in this case you are forced to resort to the sick leave mentioned above for the harm to be removed.
Allaah Knows best.