The word "noștri" means "our", therefore it has to be "We came with our parents" because "I came with our parents" sounds weird in English
Situation: Your brothers and sisters are sitting at home and you arrive home after picking up your parents from the old folks home (who are still in the driveway arguing with one another). Your siblings see you enter the house alone and ask, "Did you come alone? " You answer, "No, I came with our parents." Sounds weird to you?
As I understand it now, you just have to understand the context. For example, "I read it" could be interpreted in present or past tense; the reader/listener has to figure out what is meant by the surrounding context.
In that case, it's strange that Duolingo marks one construction correct but the other wrong.