Spiking Neural Networks (SNNs) offer a brain-inspired path toward highly efficient computation, but their practical deployment is constrained by the challenge of managing and executing their massive parallelism on physical hardware. This problem mirrors the historical challenge in processor design of moving beyond serial execution, a barrier broken by superscalar architectures that dispatch multiple instructions to parallel functional units. Drawing inspiration from this paradigm, we introduce a hardware-software co-design framework that treats synaptic events as parallelizable micro-operations. We present SupraSNN, a superscalar-inspired architecture that achieves high synapse-level parallelism by physically decoupling synaptic and neuronal computations. Within this architecture, a Multi-Cast Tree routes spike data to multiple parallel Synapse Processing Units serve as the computational pipelines, while a Merge Tree consolidates distributed results for processing by a unified Neuron Unit--deliberately centralizing complex neuron state dynamics to mitigate hardware overhead and resource duplication. The efficacy of this architecture is enabled by a sophisticated partitioning and scheduling framework that first maps the SNN onto hardware respecting memory constraints, then heuristic scheduling determines the synaptic execution order, maximizing throughput and resource utilization. Implementing a feedforward SNN trained on MNIST (93.44% accuracy), SupraSNN achieves 149 $μs$ inference latency and 0.025 mJ per image (0.276 nJ per synapse) on the Xilinx Zynq XC7Z020 FPGA--delivering 47.6% lower latency and 5.6$\times$ better energy efficiency than prior FPGA-based SNN accelerators. Beyond vision tasks, a recurrent SNN on the Spiking Heidelberg Dataset (71.82% accuracy) achieves 1.41 ms latency and 0.77 mJ per sample on XC7Z030.