This phase I trial investigates the side effects and best dose of talazoparib when given together with trifluridine/tipiracil for the treatment of patients with colorectal or gastroesophageal cancer that has spread to nearby tissue or lymph nodes (locally advanced) or other places in the body (metastatic). Drugs used in the chemotherapy, such as trifluridine/tipiracil, work in different ways to stop the growth of tumor cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Talazoparib may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth. Giving talazoparib with trifluridine/ tipiracil may inhibit certain enzymes in the cells that are responsible for tumor cell growth.

PRIMARY OBJECTIVE:

I. To determine the safety, maximum tolerated dose (MTD), and recommended phase 2 dose (RP2D) of trifluridine and tipiracil hydrochloride (trifluridine/tipiracil [FTD/TPI]) in combination with talazoparib tosylate (talazoparib) in patients with advanced colorectal (CRC) or gastroesophageal (EGC) adenocarcinoma.

SECONDARY OBJECTIVES:

I. To determine the pharmacokinetics (PK) and pharmacodynamic (PD) markers of activity.

II. To evaluate the preliminary antineoplastic efficacy of the combination.

Inclusion Criteria:

• Histologically or cytologically confirmed CRC or EGC adenocarcinoma that is locally advanced or metastatic (Cohort A); histologically or cytologically confirmed p53mt/RASonc (Cohort B1) or p53mt/RASwt CRC (Cohort B2) that is locally advanced or metastatic. Patients with adenocarcinoma histology only are allowed to participate.
• Has received at least one prior line of therapy with progression or intolerance
• Have measurable disease per Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 criteria present
• Have an Eastern Cooperative Oncology Group (ECOG) performance status of 0-1
• Life expectancy >= 3 months by investigator assessment
• Hemoglobin >= 9 g/dL
• Absolute neutrophil count >= 1500/mm^3
• Platelet count >= 100,000/mm^3 without transfusion or growth factor support
• Creatinine < 1.5 upper limit of normal (ULN) or creatinine clearance > 60 mL/min
• Total bilirubin < 1.5 x ULN
• Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) =< 2.5 x ULN or < x 5 ULN in the presence of liver metastasis
• Albumin > 3 g/dL
• Ability to swallow oral medications
• Participants of child-bearing 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:

• Systemic antineoplastic therapy within 2 weeks prior to day -14 (Dose Escalation, Cohort A) or Cycle 1 day 1 (Dose Expansion, Cohorts B1 and B2) or within the past 6 weeks if this treatment is mitomycin C or nitrosourea
• Radiotherapy within the past 2 weeks excluding palliative radiotherapy to painful bone lesions
• Prior treatment with PARP inhibitor, FUDR or FTD/TPI
• Any condition that in the investigator's opinion can limit absorption of FTD/TPI or talazoparib from the gastrointestinal (GI) tract
• Gastrointestinal obstruction (without diversion) or perforation within 4 weeks from initiation of day -14 (Dose Escalation, Cohort A) or Cycle 1 Day 1 (Dose Expansion, Cohorts B1 and B2.
• Refractory ascites (requiring weekly or more frequent paracentesis or permanent indwelling peritoneal catheter)
• Untreated central nervous system (CNS) disease. Patients with leptomeningeal disease are ineligible but patients with treated, stable CNS metastasis for at least 4 weeks are allowed to participate
• Significant cardiac disease defined as congestive heart failure stage III or IV (New York Heart Association [NYHA]), acute coronary event, cerebrovascular event, peripheral arterial embolic event, venous thromboembolic event (pulmonary embolism or lower extremity deep vein thrombosis), or ventricular arrhythmia within the past 3 months
• Other malignancy requiring active therapy
• Presence of toxicities from prior therapy of grade 2 or higher
• Active infection requiring antibiotic therapy
• Known human immunodeficiency virus (HIV) or hepatitis B infection or untreated hepatitis C infection. Patients with treated hepatitis C infection and undetectable viral load are allowed to participate
• Any history of myelodysplastic syndrome, acute leukemia, or bone marrow transplant
• Pregnant or nursing female participants
• 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