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.
I wrote "They are wandering with their parents." and got it wrong. Why? I've sometimes got it marked wrong for using "gehen" or "laufen" when it should have been "spazieren". I think in that case it even was "spazieren" which was only accepted as "to wander", but here it's wrong. Can anyone explain if it's because of DL or something about the actual German grammar?