Yes, Hungarian has this kind of emphasis built in. See https://en.wikipedia.org/wiki/Hungarian_grammar#Emphasis for examples of how varying the word order can change the emphasis and thus effectively the meaning of a sentence.

Because it really affects the meaning, I also think that your parent's statement that "you typically rearrange the sentence to emphasize the important part" is not strong enough. You don't just "typically" do this. You do it because you must, because most permutations of a given set of words -- although perfectly grammatical, and referring to the same event -- are simply pragmatically incorrect in the context of the conversation or text.