The three couples were all at the clinic on the same day in August 2018 to have embryos transferred. That implantation did not result in a successful pregnancy, the court papers say.
The Californian couple have now taken custody of their baby. The twins actually belonged to two other couples who were also clients of the hospital.
A bungled IVF treatment - which led to a NY woman unknowingly giving birth to the babies of two couples who were complete strangers - has been branded the "worst embryo-related tragedies in history".
"What Anni and Ashot discovered, much to their horror, was that their son had been stolen from them when he was still an embryo and implanted into a stranger that later became his birth mother", according to the lawsuit filed Wednesday in Los Angeles Superior Court against CHA.
"She said, 'Think of it as a good thing".
Adam Wolf, an attorney representing the couple, told AFP that the mix-up was "one of the worst fertility center tragedies I have ever seen". "You have a son now.' And I just lashed out", Anni said.More news: Sagan wins fifth stage, Alaphilippe retains lead | Sports , Other Sports
Wolf, the couple's lawyer, said the Manukyans were seeking emotional and punitive damages as well as court costs in their fight for their infant son in NY and payments made to CHA. The Queens couple was then forced to give up the babies to their genetic parents.
In vitro fertilisation is a series of procedures in which an egg is fertilised in a lab and then transferred to a uterus, according to the Mayo Clinic.
CHA Fertility claimed on its website to be one of the premier fertility treatment networks in the world and that it has "fulfilled the dreams of tens of thousands of aspiring parents" in over 22 countries. On this second attempt, though, they said the fertility clinic bungled their IVF process and have not been forthcoming since then.
CHA declined to comment after the first lawsuit was filed, and could not be reached for comment on the Manukyans' suit.
Anni and Ashot Manukyan, of Glendale, are suing Los Angeles CHA Fertility Center after the clinic placed two embryos - one belonging to them and one to another couple - into a woman from NY.
"CHA inserted a stranger's embryo into my womb, leaving us feeling violated by people that we trusted", Anni Manukyan said.
The next several weeks were a "non-stop nightmare" as they tried to get more information on their genetic son, the lawsuit states.More news: Celtics waive fan favorite which may open door for Tacko Fall
The couple said they were granted custody in May and saw their baby boy for the first time, six weeks after he was born. "It is clearly one of the worst fertility-center tragedies in US history". Who wants to meet their child in the lobby of a hotel? "It was heartbreaking. It was bad".
"It was hard not knowing where your child is", Anni said.
"He's had no issues bonding with us".
The woman gave birth to twins in March, only for DNA tests to allegedly show the babies were not related to her or even to each other.
"I wasn't there for his birth, I did not carry him, I did not feel him kick inside of me", she told Today. "She's a wonderful woman and I pray for her every day", she said. "He's been ours since day one".
The "heartbreaking" mix-up originally occurred in August a year ago when the embryos of three separate couples, who were all being treated at the Los Angeles CHA Fertility Center on the same day, were implanted into the wrong women.
To make matters worse, Anni Manukyan later learned she had been implanted with an embryo belonging to yet another couple who were clients of the clinic - but CHA has no idea who.More news: Study finds possible link between sugary drinks and cancer
"It means that we live with the uncertainty that another embryo of ours may be born to someone else", Anni Manukyan said.