ESIC Referral and Reimbursement Process
ESIC has its own referral and reimbursement system. Copies of a few instructions issued by ESIC in the past (which are within my knowledge) are uploaded below. I believe you should review them to understand the procedures you need to follow and how to comply with the instructions, making it easier for you to receive reimbursement.
In summary, the ESIS dispensary to which you are attached is the appropriate office from which you should obtain a referral for your father to any private hospital in case of an emergency. I recommend that you contact the ESIS dispensary with your ESIC Identity Card/Family Identity Card and treatment papers from the private hospital. Additionally, you can verify your eligibility by discussing the matter with the Branch Office of ESIC to which you are attached.
Best wishes.