。
The parents didn't know what their daughter wanted.
Literal
Parents [topic-は] daughter [subject-が] what [object-を] is-wanting [embedded-question-のか] didn't-understand.
欲しがっている is the third-person desire form: 欲しい becomes 欲しがる when describing someone else's wants — Japanese grammar requires this shift because emotions are directly accessible only for the speaker. 何を欲しがっているのか is an embedded question functioning as the object of わかる.