All I wanted was what every Jewish mother wants, for you to marry a doctor!— Sharon Quinzel
Sharon married Nicholas Quinzel at an unknown age. They had two children, a daughter named Harleen and a son named Barry. Sharon always wanted Harley to marry a doctor, and was not satisfied with her becoming one (although this might actually be because Harley doesn’t realize she’s not a doctor anymore due to her losing her psychiatric license because she became Harley Quinn) At some point in time, Barry died and Nicholas was imprisoned due to nearly murdering an Irishman.
Sharon became a pariah in her neighborhood after her daughter became Harley Quinn due to having a supervillain for a daughter. At some point, Harley broke contact with her for three years. During this period, Nicholas was released and they got back together. When they learned that a million-dollar bounty had been placed on Harley, Nick suggested killing their own daughter to obtain the money. Due to the fact that she was disappointed in Harley for becoming a supervillain and not marrying a doctor, Sharon agreed to allow Nick to do this.
Potentially both of her and her husband are sociopaths. Nick more obviously since he is a conman who puts his own skin before his daughter's dreams and even tries to have her killed. While Sharon is nicer, she drops the facade once Harley fails to die after the second assassination attempt and seems to have a very controlling view of her daughter's life. Both put on the mask of caring about Harley while still secretly plotting to have her killed to collect on the bounty for her. Also, neither of them seems particularly perturbed by their own parents or parents in law getting killed in front of them.