Richard's father was born Elliott Liebman in Hoboken, New Jersey on July 16, 1893. Both his parents, Philip and Rosy, were Jewish and were listed in the 1900 census as having been born in Romania. They are buried in the Har Nebo Cemetery:
https://www.findagrave.com/memorial/219440973/philip_liebman
At some point Elliott added Lester to his name. His draft registration card for World War I was signed Elliott L. Liebman. He was still using that name on January 16, 1924, when he gave an address on Eugene O'Neill to the Society Of Arts and Letters in Philadelphia, where he was teaching English at Central High School. On December 19, 1921, Liebman had given a lecture at the Young Women's Hebrew Association. By the time he was drafted again for World War II, however, he was signing his name as Elliott Lester, the same name he had used to write successful plays since, at the latest, 1925, when he debuted with The Mud Turtle:
https://www.ibdb.com/broadway-cast-staff/elliott-lester-5365
Richard's mother was born Ella Young in Philadelphia in 1893. Her father, William Young, was born in Artnagross, Antrim, Northern Ireland in 1854 and had emigrated to the US in 1873 (sources: 1900, 1920 and 1930 censuses). Her mother, Ella Coombs Flinn, had been born in Delaware. Although Richard's mother appears not to have been Jewish, she received her occupational qualification from the Jewish School of Nursing.
Richard Lester was born in 1932 and we may infer that his father had long-ceased using the name Liebman by that point. I have not (yet) seen a birth record that can confirm whether Liebman was part of his legal birth name. What is known, however, is that his parents had a stillborn baby and that only the surname Lester appears on the death certificate, applied to the baby and the parents (note that the stillbirth took place at the Jewish hospital and the baby was cremated here):
I infer that Richard's mother Ella took the married name Lester not Liebman when they wed in the 1920s, perhaps because she brought a daughter with her and they wanted the daughter to be a Lester not a Liebman. It logically follows that Richard would have been born Richard Lester and not Richard Lester Liebman, and that would be my working assumption unless other information emerges.
Another factor is that Elliott and Ella were atheists and that Richard himself is a strident atheist. Lester's biographer, Andrew Yule, claims that his mother became an atheist after being abandoned by her first husband (p.22). However, Yule's book gets Lester's mother's birthplace wrong and has her father arriving two decades later than he actually did, so its reliability is suspect. The biography fails to mention his father's Jewish ancestry or birth name or the fact his mother had graduated from a Jewish school.