Why do you insist on paying your own tuition when your parents are happy to help you financially?
Literal
Parents [subject-が] gladly financially support [for-you-てくれる] [despite-のに], why you [topic-は] own-self [by-で] tuition [object-を] pay [nominalizer-こと] [to-に] insist [question-のか].
のに (despite/even though) marks the contrast between the parents' willingness and the student's stubbornness. 固執する (to insist/persist) is a strong word for stubborn attachment. 学費 (tuition) is a compound: 学 (study) + 費 (expense). なぜ (why) opens the rhetorical question.