Nassau Co is notorious for being Anti 2A. Several months waiting for the permit is the norm. Then when your dad decides what gun he wants he will have to pay for it then submit the receipt to the county to add the gun to his permit. This could take several more months.
Once it's added to his permit he will be able to pick up his property.
Your dad is also correct that he cannot even touch a handgun in NY until he has his permit. The exception is if he is at a range being instructed by a NRA certified instructor. The county that issued my permit requires a safety course be taken prior to receiving your permit. The one I took had a live fire portion. .22 LR but it's good for new shooters.
The really twisted thing about NY is that permits are issued by the county so if you are in a Pro 2A county like where I got my permit you are issued a full carry permit. I can carry in every county in NY except for the 5 boroughs. Your dad may get a sportsman's permit from Nassau Co which will allow him to carry while Hiking, Hunting, Fishing, or Camping. Some counties are restrictive about issuing those and I would imagine Nassau is one of them.
Best of luck to him.