Joint bank account upon death

Hi, my mother recently passed in 2019 and the money in our joint savings account passed to me.   My mother put in all the money and my name was added to help her manage her bills and with the intention avoided probate. The amount was over 15,000 annual gift exclusion amount. How should I treat this money after her death? I was told no need to report on my personal tax return but want to confirm if my father, whom needs to file their last joint tax return, needs to file any gift or estate returns using form 709 or 706. 

 

Will someone please point me in the right direction?  Thank you.