Decision Transformers (DTs) effectively capture long-range dependencies using self-attention but struggle with fine-grained local relationships, especially the Markovian properties in many offline-RL datasets. Conversely, Decision Convformer (DC) utilizes convolutional filters for capturing local patterns but shows limitations in tasks demanding long-term dependencies, such as Maze2d. To address these limitations and leverage both strengths, we propose the Long-Short Decision Transformer (LSDT), a general-purpose architecture to effectively capture global and local dependencies across two specialized parallel branches (self-attention and convolution). We explore how these branches complement each other by modeling various ranged dependencies across different environments, and compare it against other baselines. Experimental results demonstrate our LSDT achieves state-of-the-art performance and notable gains over the standard DT in D4RL offline RL benchmark. Leveraging the parallel architecture, LSDT performs consistently on diverse datasets, including Markovian and non-Markovian. We also demonstrate the flexibility of LSDT's architecture, where its specialized branches can be replaced or integrated into models like DC to improve their performance in capturing diverse dependencies. Finally, we also highlight the role of goal states in improving decision-making for goal-reaching tasks like Antmaze.