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The usage of different technologies and smart devices helps people to get medical services remotely for multiple benefits. Thus, critical and sensitive data is exchanged between a user and a doctor. When health data is transmitted over a common channel, it becomes essential to preserve various privacy and security properties in the system. Further, the number of users for remote services is increasing day-by-day exponentially, and thus, it is not adequate to deal with all users using the one server due to the verification overhead, server failure, and scalability issues. Thus, researchers proposed various authentication protocols for multi-server architecture, but most of them are vulnerable to different security attacks and require high computational resources during the implementation. To Tackle privacy and security issues using less computational resources, we propose a privacy-preserving mutual authentication and key agreement protocol for a multiserver healthcare system. We discuss the proposed scheme's security analysis and performance results to understand its security strengths and the computational resource requirement, respectively. Further, we do the comparison of security and performance results with recent relevant authentication protocols.