The complexity and cost of training machine learning models have made cloud-based machine learning as a service (MLaaS) attractive for businesses and researchers. MLaaS eliminates the need for in-house expertise by providing pre-built models and infrastructure. However, it raises data privacy and model security concerns, especially in medical fields like protein fold recognition. We propose a secure three-party computation-based MLaaS solution for privacy-preserving protein fold recognition, protecting both sequence and model privacy. Our efficient private building blocks enable complex operations privately, including addition, multiplication, multiplexer with a different methodology, most-significant bit, modulus conversion, and exact exponential operations. We demonstrate our privacy-preserving recurrent kernel network (RKN) solution, showing that it matches the performance of non-private models. Our scalability analysis indicates linear scalability with RKN parameters, making it viable for real-world deployment. This solution holds promise for converting other medical domain machine learning algorithms to privacy-preserving MLaaS using our building blocks.