I think "ihr" could mean "her" or "their" in either case. The sentence above can mean "They are walking with her/their parents.
Similarly, if the first two words were,"Sie spaziert..." you could translate as, "She is walking with her parents" or as "She is walking with their parents" - but if you mean "their" you'd probably need to clarify that elsewhere as the reader might otherwise assume "her" after "She".
I understand that Sie means "they" in this sentence and is capitalized because it is at the beginning of the sentence and ihren (without being capitalized) should mean "their" from a logical standpoint. I don't understand WHY they would translate as "her" parents. It does not make any sense to me. Sorry, Duolingo.
No, spazieren gehen sounds more like a construction you'd use if asking someone if they'd like to go for a stroll. "Möchstest du spazieren gehen?" But if you get a call while strolling and someone wants to know what you're doing, you'd just say, "Wir spazieren." That's how I understand it, at least. I'm not sure if you can say, "Wir gehen spazieren" — maybe it would work if you were just on the way out the door at the time.