Accurate assessment of burn severity is crucial for the management of burn injuries. Currently, clinicians mainly rely on visual inspection to assess burns, characterized by notable inter-observer discrepancies. In this study, we introduce an innovative analysis platform using color burn wound images for automatic burn severity assessment. To do this, we propose a novel joint-task deep learning model, which is capable of simultaneously segmenting both burn regions and body parts, the two crucial components in calculating the percentage of total body surface area (%TBSA). Asymmetric attention mechanism is introduced, allowing attention guidance from the body part segmentation task to the burn region segmentation task. A user-friendly mobile application is developed to facilitate a fast assessment of burn severity at clinical settings. The proposed framework was evaluated on a dataset comprising 1340 color burn wound images captured on-site at clinical settings. The average Dice coefficients for burn depth segmentation and body part segmentation are 85.12 % and 85.36 %, respectively. The R2 for %TBSA assessment is 0.9136. The source codes for the joint-task framework and the application are released on Github (https://github.com/xjtu-mia/BurnAnalysis). The proposed platform holds the potential to be widely used at clinical settings to facilitate a fast and precise burn assessment.