This phase II trial studies the effect of talazoparib with androgen deprivation therapy and abiraterone in treating castration sensitive prostate cancer patients. Talazoparib is an inhibitor of PARP, an enzyme that helps repair deoxyribonucleic acid (DNA) when it becomes damaged. Blocking PARP may help keep tumor cells from repairing their damaged DNA, causing them to die. PARP inhibitors are a type of targeted therapy. Androgen can cause the growth of prostate tumor cells. Degarelix, leuprolide acetate, bicalutamide, goserelin acetate, and abiraterone lowers the amount of androgen made by the body. This may help stop the growth of tumor cells that need androgen to grow. Giving talazoparib with androgen deprivation therapy and abiraterone may improve cancer control for patients with castration sensitive prostate cancer.