This work focuses on the efficient numerical solution of large, stiff initial value problems (IVPs) that arise from the spatial discretization of nonlinear advection-diffusion-reaction partial differential equations (PDEs). To tackle these problems, we introduce a new class of linearly implicit two-step peer methods that incorporate tailored preconditioners, based in TASE operators [2, 3, 4, 5], and reuse previously computed stages [1] to enhance performance. The proposed methods are designed to exhibit strong stability properties—specifically L-stability or L(t)-stability with t approaching 90 degrees—while maintaining low error constants. In comparison to recently developed linearly implicit peer schemes, the new methods significantly reduce both the number of function evaluations and the number of linear systems solved per time step, while also achieving substantially lower error constants. Numerical experiments on nonlinear advection-diffusion-reaction problems from a variety of application domains demonstrate the methods' efficiency, accuracy, and robust stability. [1] M. Calvo, J. I. Montijano, L. Rández, and Saenz de-la Torre A. Explicit two-step peer methods with reused stages. Appl. Numer. Math., 195:75--88, 2024. [2] L. Aceto, D. Conte, and G. Pagano. On a generalization of time-accurate and highly stable explicit operators for stiff problems. Appl. Numer. Math., 200:2--17, 2024. [3] L. Aceto, D. Conte, and G. Pagano. Modified TASE Runge-Kutta methods for integrating stiff differential equations. SIAM J. Sci. Comput., 2025. [4] M. Bassenne, L. Fu, and A. Mani. Time–accurate and highly–stable explicit operators for stiff differential equations. J. Comput. Phys., 424, 2021. [5] M. Calvo, J. I. Montijano, and L. Rández. Modified singly–Runge–Hutta–tase methods for the numerical solution of stiff differential equations. J. Sci. Comput., 103(1)(3), 2025.