"Where are your parents?"
"Note that お o- is usually only applied to words of Japanese origin, and most Sino-Japanese words take the honorific prefix 御(ご) instead, as in 御主人(ごしゅじん) master/husband. [元気(げんき) genki is a notable exception.]"
You can start the sentence with あなたのりょうしん (your parents) but the Japanese language is terse by default, so things that can be inferred from context are often omitted such as starting a sentence with 私は (me) or 私の (my). No context is really given here, so by itself this sentence could even mean "Where are my parents." But I think the point they're trying to make is, you can usually omit that leading qualifier "私は" or "私の" or "あなたは" or "あなたの".
The particle is not specified. Duolingo would like us to translate every unspecified particle as "having to do with the listener", ie, "your parents". however, this could be said by a child who has lost their parents with the meaning of "where are my parents", but this is determined by context which is not yet provided in the Duolingo course.
You're right - you haven't used a conjunction. If you used 父and 母over 両親you would need to use a conjunction - と to make 父and 母 into a joint subject. Also, Duo is asking us for the Japanese word for parents - not the separate words for mother and father, and you would never use 父 and 母 to refer to someone else's parents.