"My parents are from Tokyo."
So does one never honor one's own parents by using the prefix? Is it because you are close you don't use keigo?
ご両親/両親 is similar to usage of お父さん/父 and お母さん/母. You are respectful to your parents and use お父さん/お母さん when talking to them but use 父/母 when talking about them to someone else. It's not polite to use honorifics when talking about your own "in-group" with someone else.
When using honorific speech you should use honorifics when referring to the listener and their group but use humble speech when referring to your own actions or group. Since ご is an honorific prefix you only use it when talking about someone else's family, not yours.
Is it correct to leave out "watashi no"? Does "ryoushin" by itself indicate that they are my parents (as it's not "goryoushin")?
As long as it's understood in the conversation that you're talking about your parents then it's absolutely fine to leave off 私の.
Oddly, I left it out and got a response suggesting "Another correct solution:" followed by the わたしの… version!
しゅっしん is a bit more posh (because it's Sino-Japanese), while から来ます is the more basic version.
In general, if you have multiple ways of expressing the same thing, kanji-heavy compounds are a fancier way of saying it than words in kana, though in some cases the kanji (even in spoken language) can be helpful in distinguishing between synonyms. In this case, 出身 indicates your origin. E.g. the place you were born/raised*, rather than where you physically just came from, which is how から is often used. Both can be used as the English "from".
*also used for school you graduated from, or field you specialized in.
Well I used 僕 instead of 私, and Duolingo cleverly marked me wrong... Gosh Duo really needs fixing on issues like this. Accepting synonyms... or the kanji form and the kana form....
僕 is specifically male. Duolingo doesn't know the answerer's sex so such isn't accepted, just as 俺は isn't
I wouldn't say "stuck up" but rather a bit brutish. It's a rough way to speak.
の indicates possession, so saying りょうしんのとうきょう would mean "my parents' Tokyo". に is not the correct particle to use with しゅっしん.
One way that you could change the word order around would be りょうしんのしゅっしんはとうきょうです (ryoushin no shusshin wa toukyou desu), which is maybe what you were going for?
When you hover over "from" it says in japanese tonode rather than shyushin. Why is this?
I put です and it's wrong. It's probably because it wasn't part of the word bank.
両親は東京終身です。 should be accepted, it is unnatural to add "I" at the start of the sentence, it is implied.
Said it 両親の出身は東京です。Why is this wrong? I really wish they'd give app users some content so we no when to leave off 私 . :(
'わたしの（my） りょうしん（parent's' plural）' is correct order.
Ryoushin is a collective noun that refers to both parents, which I believe is what sora_Japan was helpfully pointing out.