Currently, state-of-the-art RL methods excel in single-task settings, but they still struggle to generalize across multiple tasks due to catastrophic forgetting challenges, where previously learned tasks are forgotten as new tasks are introduced. This multi-task learning capability is significantly important for generalist agents, where adaptation features are highly required (e.g., autonomous robots). On the other hand, Spiking Neural Networks (SNNs) have emerged as alternative energy-efficient neural network algorithms due to their sparse spike-based operations. Toward this, we propose MTSpark, a novel methodology to enable multi-task RL using spiking networks. Specifically, MTSpark develops a Deep Spiking Q-Network (DSQN) with active dendrites and dueling structure by leveraging task-specific context signals. Specifically, each neuron computes task-dependent activations that dynamically modulate inputs, forming specialized sub-networks for each task. Moreover, this bioplausible network model also benefits from SNNs, enhancing energy efficiency and making the model suitable for hardware implementation. Experimental results show that, our MTSpark effectively learns multiple tasks with higher performance compared to the state-of-the-art. Specifically, MTSpark successfully achieves high score in three Atari games (i.e., Pong: -5.4, Breakout: 0.6, and Enduro: 371.2), reaching human-level performance (i.e., Pong: -3, Breakout: 31, and Enduro: 368), where state-of-the-art struggle to achieve. In addition, our MTSpark also shows better accuracy in image classification tasks than the state-of-the-art. These results highlight the potential of our MTSpark methodology to develop generalist agents that can learn multiple tasks by leveraging both RL and SNN concepts.