Do you say my mom and dad's store or my mom's and dad's store

The answer

Moms and dad store

My mom and dad's store. If the two nouns own the possessed item together only add 's after the second owner. example (I went to visit my friends Mary and Joe's house) the house is owned by both of them.

If the two nouns own two separate instance of the same possessed item and add's after both owners. example (I was always jealous of my sister's and brother's bikes) there are two bikes one owned by the brother and one owned by the sister