I am sorry about your loss.
The beneficiary designation typically lapses when the named beneficiary dies. In this case when your mother died and there was no named contingent beneficiary your father's estate became the beneficiary. I assume he left no Will, and thus his assets would pass to his heirs at law.
As the sole child you'd typically be the sole heir at law, assuming there are no children of deceased brothers or sisters. BUT there may be creditors, etc. or others with claims on the estate and thus the insurance company properly wants proof that it is okay to make payment to you.
You can go to the probate court and seek to become Administrator of your father's estate. If the amount of the policy is modest and there were nother major assets, you can use what many states offer -- a very simplified procedures to do it yourself. IF things are more complicated you'll need a lawyer.