The SSN application would give you parents names ($27?), other than that this posting is the only Ml Drzik in Chicago I'm seeing in the census (1930 - age 20??) living with the second marriage:
Submitter: Judy Fusko Subject: Looking for Arbet family Message: My husband is related to an Elizabeth Arbet from Cook County, IL. How he is related I am not yet sure, his grandmother, Elizabeth Nadovic Fusko (1870-1935) is a cousin, niece, or aunt to Elizabeth Arbet. Elizabeth Nadovic Fusko's parents are Elizabeth Drokema and Thomas Petrodic. Elizabeth Arbet was born Jan 1, 1885, probably in Czechoslovakia. She was married in Chicago to Michael Drzik (4 children) and then to George Pavlini (4 more children). Her father's name may have been John Arbet. She might have siblings - Anna, John, and Katherine.