Inclusion Criteria:
* Willing and able to provide informed consent.
* Cytologically or histologically confirmed advanced and incurable solid malignancy.
* For the Phase 1b, Part 2 cohort, participants must have a tumor type for which atezolizumab is an Food and Drug Administration (FDA) approved therapy.
* Eastern Cooperative Oncology Group (ECOG) performance status of =\< 1.
* Able to swallow the study drugs, has no known intolerance of study drugs or excipients, and able to comply with study requirements.
* Absolute neutrophil count (ANC) \>= 1,500 /mcL.
* Platelets \>= 100,000 / mcL.
* Hemoglobin \>= 9 g/dL or \>= 5.6 mmol/L without transfusion or erythropoietin (EPO) dependency (within 4 weeks of first dose).
* Serum creatinine =\< 1.5 X upper limit of normal (ULN) OR measured or calculated creatinine clearance \>= 30 mL/min for participants with creatinine levels \> 1.5 X institutional ULN. (Glomerular filtration rate \[GFR\] can also be used in place of creatinine or creatinine clearance \[CrCl\]).
* Creatinine clearance should be calculated using the standard Cockcroft and Gault equation.
* Serum total bilirubin =\< 1.5 X ULN OR direct bilirubin =\< ULN.
* Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\]) and alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 X ULN OR =\< 5 X ULN for participants with liver metastases.
* Albumin \>= 2.2 mg/dL.
* International Normalized Ratio (INR) or Prothrombin Time (PT) =\< 1.5 X ULN unless participant is receiving anticoagulant therapy, and then only as long as PT or partial thromboplastin time (PTT) is within therapeutic range of intended use of anticoagulants.
* Activated Partial Thromboplastin Time (aPTT) =\< 1.5 X ULN unless participant is receiving anticoagulant therapy, and then only as long as PT or PTT is within therapeutic range of intended use of anticoagulants.
* Female participants of childbearing potential must have a negative urine or serum pregnancy test. If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required. The serum pregnancy test must be negative for the participant to be eligible, and participants must agree to use a highly-effective birth control method from the time of the first study drug treatment through 180 days after the last study drug treatment, or be of nonchildbearing potential. Nonchildbearing potential is defined as follows (by other than medical reasons):
* \>= 45 years of age and has not had menses for \> 1 year
* Participants who have been amenorrhoeic for \< 2 years without history of a hysterectomy and oophorectomy must have a follicle stimulating hormone value in the postmenopausal range upon screening evaluation
* Post-hysterectomy, post-bilateral oophorectomy, or post-tubal ligation. Documented hysterectomy or oophorectomy must be confirmed with medical records of the actual procedure or confirmed by an ultrasound. Tubal ligation must be confirmed with medical records of the actual procedure.
* Male participants 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 study-drug treatment through 180 days after the last study drug treatment. Contraception should be considered for a non-pregnant female partner of childbearing potential.
* Male and female participants must agree not to donate sperm or eggs, respectively, from the first study-drug treatment through 180 days after the last study drug treatment.
* Female participants must agree to not breastfeed during the study or for 180 days after the last dose of study treatment.
* Participants must agree to not donate blood during the study or for 90 days after the last dose of study treatment.
* FOR PHASE 2 ONLY: Cytologically or histologically confirmed advanced and incurable solid malignancy. For the randomized phase 2 portion of the trial, cytologically or histologically confirmed small cell lung carcinoma (SCLC) with extensive-stage disease is required.
* FOR PHASE 2 ONLY: Complete response (CR) or partial response (PR) (per RECIST 1.1) following 4 to 6 cycles of platinum-based chemotherapy.
* Thoracic irradiation received as part of the treatment regimen and prophylactic cranial irradiation with a washout period of 14 days are allowed. Participants with limited stage disease receiving thoracic irradiation are excluded.
* FOR PHASE 2 ONLY: Able to proceed to randomization within 7 weeks after day 1 of the last cycle of prior chemotherapy.
Exclusion Criteria:
* Has not recovered (recovery is defined as National Cancer Institute Common Terminology Criteria for Adverse Events \[CTCAE\], version 4.0, grade =\< 1) from the acute toxicities of previous therapy, except treatment-related alopecia or laboratory abnormalities otherwise meeting eligibility requirements.
* Use of antineoplastic therapies within 21 days before day 1 of study treatment. (Atezolizumab does not require a washout.) Use of prophylactic cranial irradiation or thoracic irradiation within 14 days before day 1 of study treatment. Palliative radiation to bone lesions must be completed at least 7 days before day 1 of study treatment.
* Use of any other investigational agent within 21 days before day 1 of study treatment.
* Progressive or symptomatic brain metastases. Brain metastases that have been radiated, are asymptomatic, and on a stable or decreasing dose of steroids are allowed. Leptomeningeal disease is excluded.
* Serious accompanying disorder or impaired organ function, including the following:
* Cardiac (within 3 months before randomization): any unstable ischemic disease, heart failure, or untreated arrhythmia.
* Major surgery within 3 weeks before day 1 of study treatment.
* Requirement for IV alimentation (at the time of day 1 of study treatment).
* Known history of myelodysplastic syndrome (MDS) or acute myeloid leukemia (AML).
* History of another cancer within 3 years before day 1 of study treatment, with the exception of basal or squamous cell carcinoma of the skin that has been definitively treated. A history of other malignancies with a low risk of recurrence, including appropriately treated ductal carcinoma in situ (DCIS) of the breast and prostate cancer with a Gleason score less than or equal to 6, are also not excluded.
* Gastrointestinal disorder affecting absorption.
* Participants must not be considered a poor medical risk due to a serious, uncontrolled medical disorder, nonmalignant systemic disease, or active, uncontrolled infection. Examples include, but are not limited to, uncontrolled ventricular arrhythmia, recent (within 90 days) myocardial infarction, uncontrolled major seizure disorder, unstable spinal cord compression, superior vena cava syndrome, or any psychiatric disorder that prohibits obtaining informed consent.
* Participants must not have received colony stimulating factors (e.g., granulocyte colony-stimulating factor, granulocyte macrophage colony stimulating factor, or recombinant erythropoietin) within 4 weeks prior initiating protocol therapy.
* Participants must not be pregnant.
* For both arms in phase 2, participants should already be receiving atezolizumab infusions and will continue to do so while on trial. There should be no contra-indication to a PD-1 or PD-L1 inhibitor in the opinion of the treating investigator. This includes active autoimmune disease or a chronic medical condition that requires chronic steroid therapy or immunosuppressive medication, above a dose equivalent to 10 mg prednisone. Exceptions include participants with vitiligo, resolved childhood asthma/atopy, participants who require intermittent use of bronchodilators or local steroid injections, and participants with a history of hypothyroidism or adrenal insufficiency taking a stable dose of replacement therapy.
* FOR PHASE 2 ONLY: Prior treatment with a PARP inhibitor (not including iniparib).
* FOR PHASE 2 ONLY: Participants must not have progressed following first-line chemotherapy. Participants must continue maintenance atezolizumab started during initial treatment for extensive stage (ES)-SCLC.
