Lower dose radiation therapy guided by blood tests for HPV-related throat cancer

Inclusion Criteria:

* Pathologically confirmed diagnosis of squamous cell carcinoma of the oropharynx (unknown primary, base of tongue, tonsil, oropharyngeal walls, soft palate). Cytologic or fine needle aspiration (FNA) confirmation is sufficient if a biopsy of the primary tumor is not feasible
* P16 positive immunohistochemical staining. FNA may be used as the sole diagnostic tissue. If staining was done at an outside hospital, central review by the Ohio State University (OSU) department of pathology must occur prior to trial enrollment
* Clinical stage T0, N1-N2, T1-2, N1-N2, T3-T4, N0-2 (American Joint Committee on Cancer \[AJCC\] 8th edition) including no evidence of distant metastases based on general history, imaging, physical examination, and examination with laryngopharyngoscopy
* Clinical or radiographic evidence of measurable disease at the primary site or lymph nodes. Simple tonsillectomy or excision of primary without removal of nodal disease is permitted, as is excision of gross nodes but with intact primary site
* Fludeoxyglucose F-18 (FDG) PET/CT from the base of skull to the mid-thigh is mandatory and patients cannot be enrolled without a pretreatment PET/CT. PET/CT must be completed prior to enrollment
* Pretreatment tumor tissue modified HPV virus (TTMV-HPV) particles present in plasma cell free DNA value of \>= 200 copies/mL at baseline
* Patients must provide their smoking history prior to enrollment. Patients must have =\< 10 pack years of smoking. The number of pack years will be calculated using the following formula: Frequency of smoking (cigarettes/day) x duration of cigarette smoking (years)/20
* Eastern Cooperative Oncology Group (ECOG) performance status of 0-2
* Age \>= 18
* Absolute neutrophil count: ≥ 1500/mcL (within 14 days prior to registration)
* Platelets: \>= 100,000/mcL (within 14 days prior to registration)
* Hemoglobin \>= 8.0 g/dL (use of transfusion or other intervention to achieve this is acceptable) (within 14 days prior to registration)
* Total bilirubin \>= 1.5 x institutional upper limit of normal (within 14 days prior to registration)
* Aspartate transaminase (AST) or alanine transaminase (ALT) \>= 3.0 x institutional upper limit of normal (within 14 days prior to registration)
* Serum creatinine =\< 1.5 x institutional upper limit of normal or creatinine clearance \>= 50 mL/min (Cockcroft-Gault Formula) (within 14 days prior to registration)
* Human immunodeficiency virus (HIV) infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible
* Patients with known positive hepatitis B surface antigen indicating acute or chronic infection would make patient ineligible unless viral load becomes undetectable on suppressive therapy
* Patients with history of hepatitis C virus must have been treated and cured
* For women of childbearing potential, negative serum or urine pregnancy test within 14 days of registration
* Patient or legally authorized representative must provide study specific informed consent prior to study entry

Exclusion Criteria:

* Recurrent disease
* Clinical or radiographic evidence of metastatic disease or adenopathy below the clavicles
* Cancers from an oral cavity site, even if p16 positive
* Patients with simultaneous primary cancers or separate bilateral primary tumors will be excluded, except for patients with bilateral tonsil cancers
* Prior invasive malignancy (except non-melanoma skin cancer) unless disease free for a minimum of 3 years
* Prior systemic chemotherapy or immunotherapy
* Prior radiotherapy that would result in overlap of radiation fields
* Severe active co-morbidity defined as: Unstable angina or congestive heart failure requiring hospitalization in the last 6 months. Condition requiring systemic treatment with steroids or immunosuppressive medications within 14 days of registration
* Patients with active autoimmune disease requiring systemic treatment (disease modifying agents, corticosteroids, or immunosuppressive drugs
* Patients who are pregnant, nursing, or expected to conceive or father children
* Patients who are allergic to cisplatin, carboplatin, or paclitaxel