First, please do not ignore the "What is your state" question.
Second, your father needs a new "financial advisor." Either because this one is clueless or he isn't capable of providing understandable explanations.
If your father is alive, it is a gift, not an inheritance. (If your father wishes to treat it as an advance on any inheritance in his estate planning documents, he can. But that does not mean that it is not a current gift.) A gift is not taxable income to the recipient.
Your father can give $13,000 per year to as many people as he wishes without being required to report the gift to the IRS. If he exceeds $13,000 to any person in any year, then he is required to report the gift to the IRS. That does not mean that he has to pay any gift tax since each person has a lifetime credit that takes care of any gift tax liability on the first $1 million of reportable gifts. Using that credit does cut into the available estate tax credit when he passes away. Whether that is a concern depends upon the potential size of his estate.
The only possible thing that I can think of that this "financial advisor" was referring to is:
Contributions to 529 college savings plans are considered gifts to the beneficiary of the plan. A special provision of the law allows a contributor to contribute a lump sum of 5 years worth of the annual gift allowance (currently $13,000) without having it count against the $1 million lifetime gift tax credit or the eventual estate tax credit. Such a lump sum contribution is still reportable to the IRS.
