Inclusion Criteria:

          -  Diagnosis of histologically or cytologically confirmed recurrent or metastatic HNSCC
             of the oral cavity, oropharynx, larynx, hypopharynx, or p16+ SCC of the neck (unknown
             primary).

          -  Measurable disease per RECIST. Measurable disease defined as lesions that can be
             accurately measured in at least one dimension (longest diameter to be recorded) as ≥
             10 mm with CT scan, as ≥ 20 mm by chest x-ray, or ≥ 10 mm with calipers by clinical
             exam.

          -  Incurable disease, or ineligible for (including patient declined) local therapy.

          -  At least 18 years of age.

          -  ECOG performance status ≤ 1

          -  Life expectancy ≥ 16 weeks.

          -  Normal bone marrow and organ function as defined below:

               -  Hemoglobin ≥ 10.0 g/dL

               -  Absolute neutrophil count ≥ 1,500/mcL

               -  Platelets ≥ 100,000/mcL

               -  INR or PT ≤ 1.5 x institutional upper limit of normal (IULN) and aPTT ≤ 1.5 x
                  IULN unless participant is receiving anticoagulant therapy as long as PT or aPTT
                  is within therapeutic range of intended use of anticoagulants

               -  Total bilirubin ≤ 1.5 x IULN

               -  AST(SGOT)/ALT(SGPT) ≤ 2.5 x IULN unless liver metastases are present (in which
                  case they must be ≤ 5 x IULN)

               -  Serum creatinine <1.5 x ULN or creatinine clearance ≥ 51 mL/min by
                  Cockcroft-Gault or based on a 24-hour urine test

          -  Known p16 expression, if oropharyngeal primary.

          -  Postmenopausal or evidence of non-childbearing status for women of childbearing
             potential: negative urine or serum pregnancy test within 28 days of study treatment
             and confirmed prior to treatment on Day 1.

             *Postmenopausal is defined as:

               -  Amenorrheic for 1 year or more following cessation of exogenous hormonal
                  treatments

               -  Luteinizing hormone (LH) and Follicle stimulating hormone (FSH) levels in the
                  post menopausal range for women under 50

               -  radiation-induced oophorectomy with last menses >1 year ago

               -  chemotherapy-induced menopause with >1 year interval since last menses

               -  surgical sterilization (bilateral oophorectomy or hysterectomy)

          -  Male patients must use a condom during treatment and for 3 months after the last dose
             of olaparib when having sexual intercourse with a pregnant woman or with a woman of
             childbearing potential. Female partners of male patients should also use a highly
             effective form of contraception (see Section 5.6) if they are of childbearing
             potential

          -  Ability to understand and willingness to sign an IRB approved written informed consent
             document (or that of legally authorized representative, if applicable).

        Exclusion Criteria:

          -  Progression within 6 months of curatively intended systemic therapy given for
             locoregionally advanced disease.

          -  Investigational therapy within 28 days of treatment start.

          -  Prior systemic therapy for recurrent or metastatic disease.

          -  No known CNS metastases/leptomeningeal metastatic disease.

          -  Other malignancy unless curatively treated with no evidence of disease for ≥ 2 years
             except: adequately treated non-melanoma skin cancer, curative treated in situ cancer
             of the cervix, ductal carcinoma in situ (DCIS), stage 1 grade 1 endometrial carcinoma;
             or low risk-disease per discretion of the PI.

          -  A history of allergic reactions attributed to compounds of similar chemical or
             biologic composition to olaparib, pembrolizumab, carboplatin, or other agents used in
             the study.

          -  Concomitant use of known strong CYP3A inhibitors (e.g., itraconazole, telithromycin,
             clarithromycin, protease inhibitors boosted with ritonavir or cobicistat, indinavir,
             saquinavir, nelfinavir, boceprevir, telaprevir) or moderate CYP3A inhibitors (e.g.,
             ciprofloxacin, erythromycin, diltiazem, fluconazole, verapamil). The required washout
             period prior to starting olaparib is 2 weeks.

          -  Concomitant use of known strong CYP3A inducers (e.g. phenobarbital, enzalutamide,
             phenytoin, rifampicin, rifabutin, rifapentine, carbamazepine, nevirapine, and St.
             John's wort) or moderate CYP3A inducers (e.g., bosentan, efavirenz, modafinil). The
             required washout period prior to starting olaparib is 5 weeks for enzalutamide or
             phenobarbital and 3 weeks for other agents.

          -  Resting ECG indicating uncontrolled, potentially reversible cardiac conditions as
             judged by the investigator (e.g., unstable ischemia, uncontrolled symptomatic
             arrhythmia, congestive heart failure, QTcF prolongation > 500 ms, electrolyte
             disturbances, etc.) or patients with congenital long QT syndrome.

          -  Diagnosis of myelodysplastic syndrome/acute myeloid leukemia or with features
             suggestive of MDS/AML.

          -  Major surgery within 2 weeks of starting study treatment; must have recovered from any
             effects of major surgery.

          -  Has a diagnosis of immunodeficiency or is receiving chronic systemic corticosteroid
             therapy (in doses exceeding 10 mg daily of prednisone equivalent) or any other form of
             immunosuppressive therapy within 7 days prior to the first dose of treatment.

          -  Received a live vaccine or live-attenuated vaccine within 30 days prior to the first
             dose of pembrolizumab. Administration of killed vaccines is allowed. Examples of live
             vaccines is allowed. Seasonal influenza vaccines for injection are generally killed
             virus vaccines and are allowed; however, intranasal influenza vaccines (e.g. FluMist®)
             are live attenuated vaccines and are not allowed.

          -  Considered a poor medical risk due to a serious, uncontrolled medical disorder,
             non-malignant systemic disease, or active, uncontrolled infection. Examples include,
             but are not limited to, uncontrolled ventricular arrhythmia, recent (within 3 months)
             myocardial infarction, uncontrolled major seizure disorder, unstable spinal cord
             compression, superior vena cava syndrome, extensive interstitial bilateral lung
             disease on high resolution computed tomography (HRCT) scan or any psychiatric disorder
             that prohibits obtaining informed consent.

          -  Unable to swallow orally administered medication or with gastrointestinal disorder
             likely to interfere with absorption of study medication.

          -  Pregnant and/or breastfeeding. Women of childbearing potential must have a negative
             serum pregnancy test within 72 hours of first dose of treatment.

          -  Prior organ or allogeneic stem cell transplant or double umbilical cord blood
             transplantation.

          -  Has an active autoimmune disease (i.e. rheumatoid arthritis, lupus, Sjogren's
             syndrome) that has required IV or subcutaneous systemic treatment in the past 6 months
             (excluding Rituxan). Replacement therapy (i.e. thyroxine, insulin, or physiologic
             corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is
             not considered a form of systemic treatment. Patients who are known to be
             serologically positive for HIV or have known active hepatitis B or C are ineligible.

          -  History of (non-infectious) pneumonitis/interstitial lung disease that required
             steroids or has current pneumonitis/interstitial lung disease.

          -  Has a known history of active TB (Bacillus Tuberculosis).