You're exactly right. "That" is referring to "the money" without the modifier "spent by her parents." You can not replace the "that" with an "it" because "it" can only refers to an antecedent in the sentence. "That" can refer to just a part of the antecedent (the money).
I can't think of any specific examples where you can use "that" and "it" interchangeably. When you use "that" as a pronoun, you introduce a new variation of the subject.
can you please explain why the usage of 'that' is wrong in Option B of below question.
A huge flying reptile that died out with the dinosaurs some 65 million years ago, the Quetzalcoatlus had a wingspan of 36 feet, believed to be the largest flying creature the world has ever seen.
(A) believed to be
(B) and that is believed to be
(C) and it is believed to have been
(D) which was, it is believed,
(E) which is believed to be