It is possible that a newspaper obituary for your immigrant Lithuanian ancestor or for his/ her known relative might contain the name of the ancestor's birth parish. Not everyone had an obituary (because it cost money, and for other reasons) but in my experience there usually was a newspaper obituary if the deceased person was an adult.
Of special interest would be an obituary in a Lithuanian-language newspaper, because there would be an increased probability of naming the specific place in Lithuania where the deceased had resided or been born. For more information about obituaries in the Chicago newspaper "Draugas", refer to the following page on this website:
Chicago Newspaper "Draugas" Obituaries .