While homomorphic encryption (HE) provides strong privacy protection, its high computational cost has restricted its application to simple tasks. Recently, hyperdimensional computing (HDC) applied to HE has shown promising performance for privacy-preserving machine learning (PPML). However, when applied to more realistic scenarios such as batch inference, the HDC-based HE has still very high compute time as well as high encryption and data transmission overheads. To address this problem, we propose HDC with encrypted parameters (EP-HDC), which is a novel PPML approach featuring client-side HE, i.e., inference is performed on a client using a homomorphically encrypted model. Our EP-HDC can effectively mitigate the encryption and data transmission overhead, as well as providing high scalability with many clients while providing strong protection for user data and model parameters. In addition to application examples for our client-side PPML, we also present design space exploration involving quantization, architecture, and HE-related parameters. Our experimental results using the BFV scheme and the Face/Emotion datasets demonstrate that our method can improve throughput and latency of batch inference by orders of magnitude over previous PPML methods (36.52~1068x and 6.45~733x, respectively) with less than 1% accuracy degradation.