It is in Chapter Five of this novel that we can find the answer to this question. As the health of Mr. Earnshaw begins to fail and the animosity between Hindley and Heathcliff becomes ever more evident, the curate of the parish is able to step in and make a suggestion to Mr. Earnshaw regarding the future of his son and heir that would help achieve harmony in the house through his absence:
At last, our curate... advised that the young man should be sent to college; and Mr. Earnshaw agreed, though with a heavy spirit, for he said--"Hindley was nought, and would never thrive as where he wandered."
Note the way in which Mr. Earnshaw seems to have quite an accurate, if not slightly depressing, impression of his son, recognising the limitations of his character and also foreshadowing Hindley's later decline into drink and dissipation. Thus it is the curate who recommends that Hindley goes to college, and it is Mr. Earnshaw who sends him there even though he has his doubts of the benefit of such an education for Hindley.