My dads' annuity has come to term. If he takes out a lump sum and gives me the money and it exceeds $13,000 can I open a Roth IRA to avoid paying taxes on this money?

If he is giving you a gift, that money is not taxable to you.
However, your father can not avoid paying income taxes by giving the money to you as a gift.
Also, if it's more than $13,000, he will have to file a gift tax return, but won't need to pay gift taxes if he does not reach the life time gift limit (it's 5 million I believe)