Talazoparib and Gemtuzumab Ozogamicin for the Treatment of CD33 Positive Relapsed or Refractory Acute Myeloid Leukemia

Inclusion Criteria:

* Eastern Cooperative Oncology Group performance status between 0-1
* Creatinine =\< 1.5 x upper limit of normal (ULN) or calculated creatinine clearance (Cockcroft and Gault ) \> 30 mL/min (performed on plasma unless otherwise indicated)
* Alanine aminotransferase (ALT) =\< 2.5 x ULN (performed on plasma unless otherwise indicated)
* Direct bilirubin \< 1.5 mg/dL (performed on plasma unless otherwise indicated)
* Left ventricular ejection fraction (LVEF) \>= 40% as assessed by echocardiogram (ECHO) or multiple-gated acquisition (MUGA) scan performed within 28 days of enrollment
* Diagnosis of CD33+ acute myeloid leukemia (AML) with evidence of \>= 5% myeloblasts in the bone marrow, peripheral blood, or in an extramedullary site by pathology. Any CD33 receptor expression level \> 0.01% by institute flow cytometric analysis will suffice
* Relapsed or refractory disease, defined as:

  * Any bone marrow relapse after allogeneic HSCT: subjects must be at least 1 month from HSCT at the time of screening and off immunosuppressive medication for at least 2 weeks at time of initial treatment (with the exception of low-dose steroids =\< 20 mg prednisone equivalent) and have no active graft versus (vs.) host disease (GVHD)
  * AML with no prior CR/CRi after at least 1 prior cycle of remission induction chemotherapy (i.e. cytarabine or hypomethylating based regimen(s) allowed)
  * AML recurring after prior CR/CRi, which was achieved following at least one prior chemotherapy cycle (i.e. cytarabine or hypomethylating based regimen(s) allowed)
* Peripheral white blood cell (WBC) counts \<25,000/mcL. Patients are allowed to receive hydroxyurea and/or leukapheresis to control and maintain WBC count prior to enrollment and can continue on hydroxyurea through Cycle 1 Day 28 or until first disease assessment
* Participants of childbearing potential must agree to use adequate contraceptive methods (e.g., hormonal or barrier method of birth control; abstinence) prior to study entry. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately
* Participant must understand the investigational nature of this study and sign an Independent Ethics Committee/Institutional Review Board approved written informed consent form prior to receiving any study related procedure

Exclusion Criteria:

* Acute promyelocytic leukemia (APL, FAB M3) with t(15-17) and/or evidence of promyelocytic leukemia/retinoic acid receptor alpha (PML-RAR alpha)
* Blast phase of prior chronic phase chronic myeloid leukemia with t(9;22)
* Receipt of chemotherapy (except for hydroxyurea), radiotherapy, or investigational drug therapy within 2 weeks prior to treatment on study or those who have not recovered from adverse events due to agents administered \> 2 weeks earlier
* Known active central nervous system involvement; patients who have a history of central nervous system (CNS) disease which has been effectively treated as defined by at least one negative cerebrospinal sample prior to screening are eligible
* Active uncontrolled malignancy requiring ongoing chemotherapy and/or radiation. Examples of eligible patients include individuals with a prior history of malignancy treated with curative intent with no current evidence of active disease such as:

  * Subjects with stage I breast cancer that has been completely and successfully treated, requiring no therapy or only anti-hormonal therapy
  * Subjects with T1N0M0 or T2N0M0 colorectal cancer who have been completely and successfully resected and who are disease-free for \> 2 years prior to screening
  * Subjects with indolent prostate cancer, defined as clinical stage T1 or T2a, Gleason score =\< 6, and prostate specific antigen (PSA) \< 10 ng/mL, requiring no therapy or only anti-hormonal therapy
  * Subjects with a history of basal cell or squamous cell carcinoma of the skin, or carcinoma in situ of the cervix, fully resected, and with no evidence of active disease
  * Other prior or concurrent malignancies will be considered on a case-by-case basis after discussion with the principal investigator (PI)
* Uncontrolled current medical illness including, but not limited to:

  * Severe cardiac disorder (symptomatic congestive heart failure requiring treatment, chronic unstable angina pectoris, myocardial infarction, cardiac arrhythmia, torsades de pointes,
  * Neurologic impairment (cerebrovascular accident, transient ischemic attack)
  * Severe pulmonary disorder (e.g. DLCO of 65% or less or a forced expiratory volume in 1 second \[FEV1\]) of 65% of less)
  * Moderate hepatic impairment with total bilirubin \>1.5 x upper limit of normal (ULN),
  * Eastern Cooperative Oncology (ECOG ) Performance Status ≥ 2,
  * Any other psychiatric illness/social situations that in the opinion of the investigator would limit compliance with study requirements
* Known malabsorption syndrome or other condition that may impair the absorption of the study drug and/or inability and/or unwillingness to swallow capsules
* Uncontrolled active systemic fungal, bacterial, viral, or other infection with patient still exhibiting ongoing signs and symptoms due to infection despite appropriate anti-infective therapy at time of screening
* Pregnant or breastfeeding female participants
* Known active hepatitis B, active hepatitis C, or any human immunodeficiency virus (HIV) infection at the time of screening which requires therapy
* Presence of grade II-IV acute or extensive chronic GVHD at time of screening
* Unwilling or unable to follow protocol requirements
* Any condition which in the investigator's opinion deems the participant an unsuitable candidate to receive study drug including, but not limited to, medical, psychological, familial, social or geographical considerations