TCR-α/β and CD19 Depleted Stem Cell Grafts From Haplo Donors for HSCT in Relapsed Lymphoma

Inclusion Criteria: * Participants must meet one of the following disease criteria within 24 months of registration. Salvage therapy is allowed between the participant meeting one of the below criterion and registration. Participant will be considered eligible regardless of their current disease status (i.e. complete remission, partial remission, stable disease, progressive disease) unless otherwise noted below as long as one of the below criterion has been met within the previous 24 months: * Relapsed/refractory Hodgkin lymphoma after autologous stem cell transplantation * Relapsed/refractory Hodgkin lymphoma, deemed ineligible for autologous stem cell transplantation due to refractory disease * Relapsed/refractory diffuse large B cell lymphoma after autologous stem cell transplantation (history of transformed lymphoma is acceptable). Disease must be in at least complete remission or partial remission with the use of salvage therapy before study treatment commences. * Relapsed/refractory diffuse large B cell lymphoma, deemed ineligible for autologous stem cell transplantation due to refractory disease (history of transformed lymphoma is acceptable). Disease must be in at least complete remission or partial remission with the use of salvage therapy before study treatment commences. * Relapsed/refractory T cell lymphoma relapsed after at least 1 prior line of therapy * Relapsed/refractory follicular lymphoma relapsed after at least 1 prior line of therapy * Relapsed/refractory mantle cell lymphoma relapsed after at least 1 prior line of therapy * Relapsed/refractory small lymphocytic lymphoma/chronic lymphocytic leukemia relapsed after at least 1 prior line of therapy * Relapsed/refractory non-Hodgkin Lymphoma, if not specified above, relapsed after at least 1 prior line of therapy * Karnofsky score of 60% or better ("Requires occasional assistance, but is able to care for most of his/her needs"). * Pulmonary: Carbon Monoxide Diffusion Capacity (DLCO) (corrected for hemoglobin) \> 40%; and Forced Expiratory Volume (FEV1) \> 50% * Cardiac: ejection fraction (EF) ≥ 50%. No uncontrolled angina or active cardiac symptoms consistent with congestive heart failure (class III or IV), by the New York Heart Association criteria. No symptomatic ventricular arrhythmias or ECG evidence of active ischemia. No evidence by echocardiography of severe valvular stenosis or regurgitation. * Renal: estimated glomerular filtration rate (GFR) by Modification of Diet in Renal Disease (MDRD) formula \> 40 mL/min/1.73m2 * Women of child bearing potential must have a negative serum or urine pregnancy test within 14 days prior to study registration and agree to use adequate birth control during study treatment. A female of childbearing potential (FCBP) is a sexually mature female who: 1) has not undergone a hysterectomy or bilateral oophorectomy; or 2) has not been naturally postmenopausal for at least 24 consecutive months (i.e., has had menses at any time in the preceding 24 consecutive months). * Voluntary written consent Exclusion Criteria: * Active Central nervous system (CNS) lymphoma within two weeks of registration. Patients with a history of CNS involvement must have adequate treatment as defined by at least two negative spinal fluid assessments separated by at least one week. (Otherwise Lumbar Puncture (LP) is not required if no clinical suspicion or evidence of CNS involvement.) Patients who have received cranial radiation therapy must still be eligible to receive total lymphoid irradiation to 7 Gy. * New or active infection as determined by fever, unexplained pulmonary infiltrate or sinusitis on radiographic assessment. Infections diagnosed within 4 weeks of registration must be determined to be controlled or resolving prior to treatment. * Presence of HIV, or active hepatitis A, B, or C infection * Allergy or hypersensitivity to agents used within the treatment protocol. * For an indolent lymphoma histology (follicular lymphoma, Small Lymphocytic Lymphoma/Chronic Lymphocytic Leukemia (SLL/CLL)) or mantle cell lymphoma, the patient should not have an HLA-matched sibling, who would be an eligible donor, available. * History of prior mediastinal radiation * Reported illicit drug use * Vulnerable population groups, i.e., prisoners, those lacking consent capacity, non-English speaking, illiterate, pregnant females.