• Must have signed written, informed consent.

• Males or females ≥ 18 years of age.

• 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.

• Diagnosis of the disease based on WHO 2016 (Swerdlow, 2016) criteria.

• Subjects must have measurable disease as defined by Lugano 2016 criteria (Cheson, 2016).

• 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.

• 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).

• 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).

• Must be at least 90 days since ASCT, if performed.

⁃ 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).

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

∙ Serum creatinine ≤ 1.5 mg/dL or estimated creatinine clearance ≥ 30 mL/min as calculated using the Cockcroft-Gault formula and not dialysis-dependent.

‣ 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.

⁃ 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.

⁃ Must have an ECOG performance status of 0 to 1.