Inclusion Criteria:  >= 18 years of age Histologically confirmed iNHL of any of the following subtypes recognized by the World Health Organization (WHO) classification: follicular lymphoma and marginal zone lymphoma (splenic, nodal and extranodal) Patients must meet clinical criteria for requiring treatment At least two prior systemic therapies for FL (phase 2 portion) and one prior systemic therapy for MZL. Prior autologous stem cell transplant is permitted. Prior CAR-T cell therapy is permitted. For the phase 1 portion, patients receiving one prior systemic therapy are allowed. Eastern Cooperative Oncology Group (ECOG) performance status of 0-2 Creatinine clearance >= 50 ml/min using a 24-hour creatinine clearance or estimated creatinine clearance using the Cockcroft-Gault equation Bilirubin < 1.5 x upper limit of normal (ULN) Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) < 1.5 x ULN Absolute neutrophil count (ANC) > 1000/mm^3 (without growth factor support)  Platelet > 75,000/mm^3 (without transfusion support)  Unless related to bone marrow involvement with the disease, in which case platelets must be > 50,000/mm^3 Hemoglobin >= 8 gm/dL Willing and able to participate in all required evaluations and procedures in this study protocol Ability to understand the purpose and risks of the study and provide signed and dated informed consent and authorization to use protected health information Radiographically measurable disease by computed tomography (CT) scan, defined as at least one node > 1.5 cm in size or assessable disease Woman of childbearing potential (WOCBP) who are sexually active must agree to use highly effective methods of contraception during treatment and for 2 days after the last dose of acalabrutinib, and 30 days after the last dose of duvelisib. WOCBP should have negative pregnancy test at screening and follow up throughout the study. Male subjects must agree to use highly effective methods of contraception during the study and up to 1 month after last dose of duvelisib. Male fertility may be impaired based on animal data (per duvelisib label)  Exclusion Criteria:  Prior exposure to a BCR inhibitor (e.g., BTK inhibitors, phosphoinositide-3 kinase [PI3K], or Syk inhibitors) or BCL-2 inhibitor Patients with grade 3B FL or clinical evidence of transformation to aggressive lymphoma Central nervous system (CNS) involvement Prior malignancy (or any other malignancy requiring active treatment), except for adequately treated basal cell or squamous cell skin cancer, in situ cervical cancer, or other cancer which will not limit survival to < 1 year Clinically significant cardiovascular disease such as symptomatic arrhythmias, congestive heart failure, or myocardial infarction within 6 months of screening, or any class 3 or 4 cardiac disease as defined by the New York Heart Association Functional Classification. Note: Subjects with controlled, asymptomatic atrial fibrillation can enroll in study Active bleeding or history of bleeding diathesis (e.g., hemophilia or von Willebrand disease) Uncontrolled AIHA (autoimmune hemolytic anemia) or ITP (idiopathic thrombocytopenic purpura) Requires or receiving anticoagulation with warfarin or equivalent vitamin K antagonists Prothrombin time (PT)/international normalized ratio (INR) or activated partial thromboplastin time (aPTT) (in the absence of lupus anticoagulant) > 2 x ULN Requires treatment with proton pump inhibitors (e.g., omeprazole, esomeprazole, lansoprazole, dexlansoprazole, rabeprazole, or pantoprazole). Note: Subjects receiving proton pump inhibitors who switch to H2-receptor antagonists or antacids are eligible for enrollment to this study History of significant cerebrovascular disease/event, including stroke or intracranial hemorrhage, within 6 months before the first dose of study drug Major surgical procedure within 28 days of the first dose of study drug. Note: If a subject had major surgery, they must have recovered adequately from any toxicity and/or complications from the intervention before the first dose of study drug Pregnancy or lactation, or intending to become pregnant during the study Concurrent participation in another therapeutic clinical trial Known history of infection with human immunodeficiency virus (HIV) History of progressive multifocal leukoencephalopathy Grade >= 2 toxicity (other than alopecia) continuing from prior anticancer therapy Known history of hypersensitivity or anaphylaxis to study drug(s) including active product or excipient components History or concurrent condition of interstitial lung disease of any severity and/or severely impaired lung function Prior history of drug-induced colitis or drug-induced pneumonitis History of chronic liver disease or veno-occlusive disease Ongoing treatment with chronic immunosuppressants (e.g., cyclosporine) or systemic steroids > 20 mg of prednisone (or equivalent) once daily (QD)  Uncontrolled viral, bacterial, fungal or parasitic infection that is untreated or unresponsive to antimicrobial therapy  NOTE: Subjects on antimicrobial, antifungal, or antiviral prophylaxis are not specifically excluded if all other inclusion/exclusion criteria are met Concurrent administration of medications or foods that are strong or moderate inhibitors or strong inducers of cytochrome P450 3A (CYP3A). No prior use within 2 weeks before the start of study intervention Patients with prior allogeneic transplantation Unable to receive prophylactic treatment for pneumocystis, herpes simplex virus (HSV), or herpes zoster (VZV) at screening History of tuberculosis treatment within the 2 years prior to study entry Prior surgery or gastrointestinal dysfunction that may affect drug absorption (e.g., gastric bypass surgery, gastrectomy). Subjects with clinically significant medical condition of malabsorption, inflammatory bowel disease, chronic conditions which manifest with diarrhea, refractory nausea, vomiting or any other condition that will interfere significantly with drug absorption Active cytomegalovirus (CMV) or Epstein-Barr virus (EBV) infection (i.e., subjects with detectable viral load) Administration of a live or live attenuated vaccine within 6 weeks of study entry  Infection with hepatitis B, hepatitis C  Subjects with a positive hepatitis B surface antigen (HBsAg) Subjects with a positive hepatitis B core antibody (HBcAb) must have negative hepatitis B virus (HBV) deoxyribonucleic acid (DNA) to be eligible and must be periodically monitored for HBV reactivation by institutional guidelines Subject who are hepatitis C antibody (HepcAb) positive will need to have a negative polymerase chain reaction (PCR) and must be willing to undergo DNA PCR testing during the study to be eligible Investigators who strongly believe that a positive HBcAb is false (negative hepatitis C [hep C] PCR) due to passive immunization from previous immunoglobulin infusion therapy should consider the risk-benefit for the patient given the potential for reactivation