Which one of these sentences is correct?
I don't want to be the one that have to explain to your parents why you are not going to graduate.
I don't want to be the one that has to explain to your parents why you are not going to graduate.
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The sentence has a relative clause, introduced by 'that', and the embedded clause obeys the rules of subject agreement. Since the subject of the relative clause is 'the one', which is singular, you should use 'has' to agree with it, not 'have'.
(Phil's comment about subjunctive is not a part of modern English.)