Questions regarding an assumable mortgage

We recently found a fixer-upper we wanted to live in and fix up over the next few years. We owned our own home at the time so my parents purchased the house in their name with an assumable adj rate mortgage and added my husband and myself to the title planning for us to assume the mortgage after the sale of our home. Our home just sold and we want to assume the mortgage so we can have the tax benefits, as we are paying the mortgages/taxes/insurance anyway and we want the house to be exempt from capital gains tax when we sell it and it wouldn't be if kept in my parent's name. Can anyone tell me the best way to go about assuming the mortgage, if we will have to re-pay the down payment or if my parent's down payment can remain if they want it to, and if there is any way for us to claim current mortgage payment interest on taxes since we are on the title? Please let me know if further info is needed. Thanks so much!