Hyperdimensional Computing (HDC) is a brain-inspired computing paradigm that represents and manipulates information using high-dimensional vectors, called hypervectors (HV). Traditional HDC methods, while robust to noise and inherently parallel, rely on single-pass, non-parametric training and often suffer from low accuracy. To address this, recent approaches adopt iterative training of base and class HVs, typically accelerated on GPUs. Inference, however, remains lightweight and well-suited for real-time execution. Yet, efficient HDC inference has been studied almost exclusively on specialized hardware such as FPGAs and GPUs, with limited attention to general-purpose multi-core CPUs. To address this gap, we propose ScalableHD for scalable and high-throughput HDC inference on multi-core CPUs. ScalableHD employs a two-stage pipelined execution model, where each stage is parallelized across cores and processes chunks of base and class HVs. Intermediate results are streamed between stages using a producer-consumer mechanism, enabling on-the-fly consumption and improving cache locality. To maximize performance, ScalableHD integrates memory tiling and NUMA-aware worker-to-core binding. Further, it features two execution variants tailored for small and large batch sizes, each designed to exploit compute parallelism based on workload characteristics while mitigating the memory-bound compute pattern that limits HDC inference performance on modern multi-core CPUs. ScalableHD achieves up to 10x speedup in throughput (samples per second) over state-of-the-art baselines such as TorchHD, across a diverse set of tasks ranging from human activity recognition to image classification, while preserving task accuracy. Furthermore, ScalableHD exhibits robust scalability: increasing the number of cores yields near-proportional throughput improvements.