Talazoparib Plus Irinotecan With or Without Temozolomide in Children With Refractory or Recurrent Solid Malignancies

INCLUSION CRITERIA: * Patients with refractory or recurrent solid tumors for which there is no standard therapy are eligible. Patients must have had histologic verification of malignancy at original diagnosis or at the time of relapse. * 12 months - 25 years at the time of enrollment on study. * Patients must have a BSA of ≥ 0.42 m\^2 at the time of study enrollment due to capsule strength(s). * Patients must have either measureable or evaluable disease. * Life expectancy must be at least 8 weeks. * Performance status: Karnofsky ≥ 50 for patients \> 16 years of age; Lansky ≥ 50 for patients ≤ 16 years of age. * Prior therapy: Patients who have received prior therapy with an irinotecan-based or temozolomide-based regimen are eligible. Patients who have received prior therapy with a PARP inhibitor other than talazoparib are eligible; however, patients who have progressed on a PARP inhibitor plus irinotecan regimen are not eligible. * Organ function: Must have adequate organ and bone marrow function as defined by the following parameters: * Patients with solid tumors not metastatic to bone marrow: * Peripheral absolute neutrophil count (ANC) ≥1,000/mm\^3 * Platelet count ≥ 100,000/mm\^3 (no transfusion within 7 days of enrollment) * Hemoglobin ≥ 9 g/dL (with or without support) * Patients with solid tumors metastatic to bone marrow will be eligible for study but not evaluable for hematologic toxicity. These patients must not be known to be refractory to red cell or platelet transfusions. At least 2 of every cohort of 3 patients must be evaluable for hematologic toxicity. If dose limiting hematologic toxicity is observed at any dose level, all subsequent patients enrolled must be evaluable for hematologic toxicity. * Adequate renal function defined as: Serum creatinine concentration ≤3x the institutional upper limit of normal (ULN) or creatinine clearance or radioisotope GFR ≥ 70ml/min/1.73m\^2. * Adequate liver function defined as: For Part A participants: normal liver function as defined by SGPT (ALT) concentration ≤5x the institutional ULN, a total bilirubin concentration ≤2x the institutional ULN for age, and serum albumin ≥ 2g/dL. * Adequate liver function defined as: For Part B participants: normal liver function as defined by SGPT (ALT) concentration \< 2.5 times institutional ULN, a total bilirubin concentration ≤ 1.5 times the institutional ULN for age, and serum albumin ≥ 2 g/dL. * Patients must have fully recovered from the acute toxic effects of chemotherapy, immunotherapy, surgery, or radiotherapy prior to entering this study: * Myelosuppressive chemotherapy: Patient has not received myelosuppressive chemotherapy within 3 weeks of enrollment onto this study (6 weeks if prior nitrosourea). * Hematopoietic growth factors: At least 7 days must have elapsed since the completion of therapy with a growth factor. At least 14 days must have elapsed after receiving pegfilgrastim. * Biologic (anti-neoplastic agent): At least 7 days must have elapsed since completion of therapy with a biologic agent. For agents that have known adverse events occurring beyond 7 days after administration, this period prior to enrollment must be extended beyond the time during which adverse events are known to occur. * Monoclonal antibodies: At least 3 half-lives must have elapsed since prior therapy that included a monoclonal antibody. * Radiotherapy: At least 2 weeks must have elapsed since any irradiation; at least 6 weeks must have elapsed since craniospinal RT or substantial bone marrow irradiation. * All patients and/or their parents or legally authorized representatives must sign a written informed consent. Assent will be obtained, when appropriate, according to institutional guidelines. EXCLUSION CRITERIA: * Pregnant or breastfeeding. * If sexually active and with childbearing potential, must agree to use effective method of contraception, such as intrauterine device, bilateral tubal ligation for ≥ 6 months before randomization, partner vasectomized for ≥ 6 months before randomization, and sexual abstinence when in relation to the preferred and usual lifestyle of the subject. * Male subjects must use a condom when having sex with a pregnant woman and when having sex with a woman of childbearing potential from the time of the first dose of study drug through 105 days after the last dose of study drug. * Contraception should be considered for a nonpregnant female partner of childbearing potential. * Male subjects with partners of childbearing potential must use a condom and contraception should be considered for the female partner from the time of the first dose of study drug through 105 days after the last dose of study drug. * Male subjects whose partners are pregnant should use condoms for the duration of the pregnancy. * Male and female subjects must agree not to donate sperm or eggs, respectively, from the first dose of study drug through 105 days and 45 days after the last dose of study drug, respectively (hormonal contraception is not allowed) prior to study entry, during treatment, and for 45 days after last dose of study drug. * Females considered not of childbearing potential include those who are surgically sterile (bilateral salpingectomy, bilateral oophorectomy, or hysterectomy) or who are post menopausal, defined as: \< 55 years of age with no spontaneous menses for ≥ 12 months before randomization and with a postmenopausal follicle stimulating hormone (FSH) concentration \> 30 IU/L (or meeting criteria for post-menopausal status by the local laboratory). * If females with childbearing potential, a negative serum pregnancy test at Screening and willing to have additional serum and urine pregnancy tests during the study * Note: Females without childbearing potential include those in menopause ≥2 years, with tubal ligation ≥1 year before screening, or with total hysterectomy. * Concomitant medications * Corticosteroids: Patients receiving corticosteroids that have not been on a stable or decreasing dose for at least 7 days prior to enrollment are not eligible. * Investigational drugs: Patients cannot receive other investigational drugs while on this study. * Anti-GVHD drugs post-transplant: Patients receiving cyclosporine, tacrolimus or other GVHD agents are not eligible. * Prior treatment with talazoparib. * Unable to swallow capsules. * Active, uncontrolled infection. * Prior solid organ transplant. * Prior total body irradiation (TBI). * Unwilling or unable to comply with the safety monitoring requirements of this protocol.