Well the 4R70W trans is a very good trans, very strong too.
I would take it into a dealer and get a full over diagnostic (Its about $70) and see what they say before you get the car from your parents.
Its probably a great time to flush that trans fluid too and chance the filter.
Ask the dealer how much that is and ask them if they could put a new valve body on for you.
The reason why I am suggesting a new valve body is because I am going to recommend a JMod valve body.
Its pretty much what Ford SHOULD have done and is a very popular upgrade to allow more fluid to flow through the body allowing better shifting and longer life of the trans.
Its about $119 and when you ship your old one back you get $40 dollars back.
You can buy it here.
The dealer will be able to give you full insight on the health of the car and the transmission.
Good Luck and Welcome to MustangForums.