We consider the problem of sinusoidal parameter estimation using signed observations obtained via one-bit sampling with time-varying thresholds. In a previous paper, a relaxation-based algorithm, referred to as 1bRELAX, has been proposed to iteratively maximize the likelihood function. However, 1bRELAX can only be used in applications involving a small number of sinusoids due to the time-consuming exhaustive search procedure needed in each iteration. In this paper, we present a majorization-minimization (MM) based 1bRELAX algorithm, referred to as 1bMMRELAX, to enhance the computational efficiency of 1bRELAX. Using the MM technique, 1bMMRELAX maximizes the likelihood function iteratively using simple FFT operations to reduce the computational cost of 1bRELAX while maintaining its excellent estimation accuracy. Numerical examples are presented to demonstrate the effectiveness of the proposed method.