The parents' names are not needed to search for your ancestor's birth record in Lithuania, but they are needed to be confident that you have found the correct birth record in Lithuania. There could be other persons in the parish that had the same name as your ancestor (because in those times all of the relatives lived near each other, and because some of those could have the same given name).
The names of the parents of your ancestor might be available via "family recollection" or "family papers". If not, there are only a few U.S. records where their names might be found. These are:
In U.S. records other than those prepared by U.S. Lithuanians and ship passenger lists, it is typical to find the names of the ancestor's parents listed in an Americanized form.
Discussion of results:
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