First, lets look at why 踏{ふ}み込{こ}んで isn't correct. 踏{ふ}み込{こ}んで can be used as 'step into' as in 部屋に足を踏み込む, but I usually see the case where it is used as 'to make a bold start/to dive into'.
乗{の}り出{だ}す is tricky here, because most of the usage you'll see in news articles and such is that of 'embark', or 'set out'. The other dictionary meaning doesn't really fit this question so well either, 'to lean/move pressingly forward' (the latter from goo's 4th definition:
4 からだを前にぐっと出す。「身を―・して演奏を聴く」).
Dictionaries don't always hold all the meanings of a word, especially the idiomatic ones.
So I did a google search for for 親が乗り出す, and there were a large number of hits similar to the JLPT question where I would translate 乗り出す as 'intervene in or step into ( a children's fight)" and also as a secondary meaning as "to get involved (in marital life/decisions involving their children)"
I looked at many dictionaries and couldn't find any definitions that matched this one, so perhaps it will do to know that 親が乗り出す has this idiomatic use case.
For completeness I searched for 親が踏み込む as well, and came across these kinds of sentences, which does look similar to secondary meaning of 親が乗り出す, to get involved, but is a bit different from intervening or stepping into a children's fight:
1.子どもには子どもの世界がありますし、そこに親が踏み込むとストレスになるだろうし
2.そこで聞きたいのはもしお子さんが同じようになった場合どこまで親が踏み込むべきなのでしょうか
So while it is close, I think this is a sufficient explanation of 乗り出す to answer and explain the question.
乗り出す
can mean "enter", so "the parent enters the children's fight" could work here too. According to Daijirin,踏み込む
means to step into a quagmire, boldly/daringly step forwards, trespass, delve into the finer details of something, tread on an accelerator pedal or do something boldly/daringly. I'm not entirely sure why踏み込む
doesn't work, but maybe the children aren't a danger to the parent?