私定终身

私奔。

or you can say "这是一桩没有经过父母同意的婚姻"

“私定终身”means you decide to marry the man/woman without letting your parents know it. It does not necessarily mean your parents do not approve of it. They may approve of it, or they may not. “ 同居”is the wrong word, which means a man and a woman live together without getting married. It has nothing to do with patents' approval. “自主婚姻”is not a proper word either. It means THE PERSON HIMSELF decides who to get married to, not somebody else. Actually, I cannot think of an equivalent. In Chinese we just say "他们没有经过父母同意就结婚了" （They got married without their parents' approval）

you can simply say: 同居