Her parents had been opposed for a long time, but finally allowed her to go to Europe alone.
Literal
Parents [topic-は] long time opposing [concessive-が], finally she [subject-が] Europe [to-に] alone going [nominalizer-こと] [object-を] permitted.
A complex sentence with temporal depth: 長い間反対していた (had been opposing for a long time) sets the background, ついに (finally) marks the turning point. ~ことを許す nominalizes the clause with こと and marks it as the object of 許す (to permit).