Inclusion Criteria

1. Must have signed written, informed consent.
2. Males or females ≥ 18 years of age.
3. Must have prior biopsy proven confirmed diagnosis of DLBCL NOS (including DLBCL arising from indolent lymphomas), HGBL, PMBCL,and tFL or follicular lymphoma Grade 3B.
4. Diagnosis of the disease based on WHO 2016 (Swerdlow, 2016) criteria.
5. Subjects must have measurable disease as defined by Lugano 2016 criteria (Cheson, 2016).

6. Must have relapsed/refractory disease and have received adequate prior anti-cancer therapy, as defined below:

   a. Prior systemic chemotherapy must include a line of chemoimmunotherapy that includes an anti-CD20 antibody, an anthracycline, and 1 or more of the following: i. No response to first-line therapy (primary refractory disease). Refractory disease (defined as SD, PD, PR or CR with relapse before 3 months).

   ii. Progressive disease following two or more lines of therapy. However, SD as the best response after at least 2 cycles of the last line of therapy with SD duration no longer than 6 months from the last dose of therapy is also acceptable.

   iii. Refractory post-autologous stem cell transplant (ASCT). Disease progression or relapse occurring at less than or equal to 12 months of undergoing ASCT (must have biopsy proven recurrence in relapsed patients). If salvage therapy is given post-ASCT, the patient must have had no response to or relapsed after the last line of therapy.

   iv. Refractory disease (SD, PD, PR or CR with relapse before 3 months) or relapsed disease (defined as CR/PR with relapse on, or after lasting at least 3 months but no more than 12 months), to CD20 antibody and anthracycline containing first-line therapy.

7. Must be willing to practice birth control from the time of Screening and throughout the first year of the study after P-CD19CD20-ALLO1 administration (both males and females of childbearing potential).
8. Must have a negative serum pregnancy test at Screening and a negative urine pregnancy test within 3 days prior to initiating the lymphodepletion chemotherapy regimen (females of childbearing potential).
9. Must be at least 90 days since ASCT, if performed.
10. Treatment with prior CD19 targeted therapy is allowed, provided the last dose was administered at least 90 days before the start of P-CD19CD20-ALLO1 treatment in this study. Must be at least 3 months since autologous CAR-T therapy if such therapy was administered (medical monitor must approve prior CAR-T therapy or other prior T cell targeted therapy).

11. Must have adequate vital organ function, defined as follows (or medical monitor approval):

    1. Serum creatinine ≤ 1.5 mg/dL or estimated creatinine clearance ≥ 30 mL/min as calculated using the Cockcroft-Gault formula and not dialysis-dependent.
    2. Adequate hematologic function, including:

    i. Absolute neutrophil count (ANC) ≥ 1000/μL in the absence of growth factor support (granulocyte colony stimulating factor [G-CSF] within 7 days or peg-G-CSF within 14 days) ii. Platelet count ≥ 50,000/μL in the absence of transfusion support (platelet transfusion within 7 days) iii. Hemoglobin ≥ 8 g/dL in the absence of transfusion support (red blood cell count or whole blood within 7 days) c. Aspartate transaminase (AST) and alanine aminotransferase (ALT) ≤ 3 × the upper limit of normal (ULN), and total bilirubin ≤ 2.0 mg/dL (unless there is a history of Gilbert's Syndrome in which case bilirubin levels ≤ 3 mg/dL).

    d. Left ventricular ejection fraction (LVEF) ≥ 45%. LVEF assessment must have been performed within 4 weeks of enrollment.

12. Must have recovered from toxicities due to prior therapies to Grade ≤ 2 according to the NCI CTCAE v5.0 criteria or to the subject's prior baseline.
13. Must have an ECOG performance status of 0 to 1.

Exclusion Criteria

1. Is pregnant or lactating.
2. Has inadequate venous access.
3. Has active hemolytic anemia, POEMS syndrome (polyneuropathy, organomegaly, endocrinopathy, monoclonal protein, and skin changes), disseminated intravascular coagulation, leukostasis, or amyloidosis.
4. Concurrent or previous other malignancy within 2 years of study entry, except curatively treated malignancies including basal or squamous cell skin cancer, prostate intraepithelial neoplasm, carcinoma in situ of the cervix, breast, or Bowen's disease. Patients with other curatively treated malignancies with low risk of recurrence not listed may also be considered eligible after review and approval by the Sponsor medical monitor.
5. Has active autoimmune disease, such as psoriasis, multiple sclerosis, lupus, rheumatoid arthritis, etc. (the medical monitor will determine if a disease is active and autoimmune).
6. Has a history of significant central nervous system (CNS) disease, such as stroke, epilepsy, primary CNS lymphoma, etc. (the medical monitor will determine if significant).
7. Has an active systemic infection (e.g., causing fevers or requiring antimicrobial treatment).
8. Has a history of hepatitis B virus (HBV), hepatitis C virus (HCV), human immunodeficiency virus (HIV), or human T-lymphotropic virus (HTLV) infection, or any immunodeficiency syndrome. Subjects with a history of treated hepatitis C can be enrolled if negative by hepatitis C polymerase chain reaction (PCR) on multiple occasions and with medical monitor approval.
9. Is positive for human herpes virus (HHV)-6 or HHV-7 infection by PCR at the Screening Visit (subjects may be included in the study if they are HHV-6 or HHV-7 IgG antibody-positive but PCR-negative).
10. Has New York Heart Association (NYHA) Class III or IV heart failure, unstable angina, or a history of myocardial infarction or significant arrhythmia (e.g., atrial fibrillation, sustained [> 30 seconds] ventricular tachyarrhythmias, etc.).
11. Has any psychiatric or medical disorder (e.g., cardiovascular, endocrine, renal, gastrointestinal, genitourinary, immunodeficiency or pulmonary disorder not otherwise specified) that would, in the opinion of the Investigator or medical monitor, preclude safe participation in and/or adherence to the protocol (including medical conditions or laboratory findings that indicate a significant probability of not qualifying for or being unable to undergo, LD chemotherapy and/or CAR-T cell administration).
12. Has received non-mAb anti-cancer medications within 2 weeks of the time of initiating LD chemotherapy.
13. Has received mAb therapy within 4 weeks of initiating LD chemotherapy.
14. Has received immunosuppressive medications within 2 weeks of the time of administration of P-CD19CD20-ALLO1, and/or expected to require them while on study (the medical monitor will determine if a medication is considered immunosuppressive.)
15. Has received systemic corticosteroid therapy > 5 mg/day of prednisone or equivalent dose of another corticosteroid within 1 week or 5 half-lives (whichever is shorter) of the administration of P-CD19CD20-ALLO1 or is expected to require it during the course of the study. (Topical and inhaled steroids are permitted. Systemic corticosteroids are contraindicated after receiving P-CD19CD20-ALLO1 cells outside of study-specific guidance or medical monitor approval).
16. Has CNS metastases or CNS involvement (including leptomeningeal carcinomatosis, cranial neuropathies or mass lesions, cauda equina syndrome, and spinal cord compression).
17. Has a history of severe immediate hypersensitivity reaction to any of the agents used in this study.
18. Has a history of having undergone allogeneic or xenogeneic transplant, or has undergone autologous transplantation within 90 days. Subjects with prior history of allogeneic stem cell transplant may be enrolled if they are not on immunosuppressive medications and with medical monitor approval.
19. Has received prior allogeneic genetically modified cellular therapy or was treated with experimental allogeneic cell therapy.
20. History or Grade ≥ 3 HLH/MAS or neurotoxicity with prior therapies (all symptoms of HLH/MAS, neurotoxicity, or CRS from prior therapies must be resolved at the time of enrollment).
21. Has positive DAT at Screening Visit (may be allowed with medical monitor approval).