In trapped-ion quantum information processing, interactions between spins (qubits) are mediated by collective modes of motion of an ion crystal. While there are many different experimental strategies to design such interactions, they all face both technical and fundamental limitations to the achievable coherent interaction strength. In general, obtaining strong interactions and fast gates is an ongoing challenge. Here, we extend previous work [W. Ge, B. C. Sawyer, J. W. Britton, K. Jacobs, J. J. Bollinger, and M. Foss-Feig, Phys. Rev. Lett. 122, 030501 (2019)] and present a general strategy for enhancing the interaction strengths in trapped-ion systems via parametric amplification of the ions motion. Specifically, we propose a stroboscopic protocol using alternating applications of parametric amplification and spin-motion coupling. In comparison with the previous work, we show that the current protocol can lead to larger enhancements in the coherent interaction that increase exponentially with the gate time.