Occupational driving of light-duty vehicles (LDVs) became increasingly important in parcel delivery faced with the explosive growth of e-commerce. Since musculoskeletal disorders (MSDs) represent the most reported driving-related health problem, we aimed to analyze the risk of low back pain (LBP) and upper-extremity musculoskeletal disorders (UEMSDs) associated with driving LDVs for parcel delivery. In 306 postal workers exposed to driving and 100 unexposed workers, information on occupational driving, physical/psychosocial constraints, and work organization were collected via a questionnaire. MSDs were assessed using the Nordic Questionnaire, 14 additional questions regarding LBP, and a standardized clinical examination for UEMSDs. Statistical modeling consisted of multivariable logistic regression for UEMSDs and the item response theory approach for LBP. UEMSDs were associated with the distance of rural rounds and inversely associated with urban/mixed delivery rounds. Handling heavy loads was associated with LBP, and high physical demands during delivery rounds were related to MSDs. Karasek dimensions and mobbing actions were associated with MSDs. Work recognition, driving training, using an automatic gearbox, and the utilization of additional staff during peak periods were inversely associated with MSDs. Our results suggest that the distance driven in rural settings and high physical demands were associated with MSDs, while some organizational factors could protect from MSDs.