The most current U.S. Census is from 1940. Have you found your family ancestors in that census.
Your best approach is to start with what you know and work your way back. Find them in the 1940 Census and work your way back census by census.
You also can try to get the Social security card applications which list parents names (thet are redacted if the applicant was born less than 100 years ago to protect the privacy of their 120 year old parents!).
Where in the U.S. did they settle?
Your surname is a bit similar to our Lithuanian born neighbors who were Esiukevicz/Esiukevicius.