arXiv:2610.33780v1 Announce Type: new Abstract: In this paper, a one-parameter family of second-order implicit--explicit (IMEX) multistep methods is studied for delay differential equations (DDEs). The analysis is based on the matrix problem $y'(t)=A y(t) + B y(t-\tau(t))$, where $A$ is Hermitian negative definite but $B$ is any matrix. For constant, grid-aligned delays, a field-of-values argument yields unconditional stability whenever a suitable generalized field of values $F_p(A,B)$ is contained in the open unit disk. For a constant non-grid-aligned delay, we show that quadratic interpolation does not reduce the corresponding stability disk. For genuinely variable delays, the method is formulated as a nonautonomous augmented system, and product-level sufficient conditions are obtained from a common quadratic Lyapunov inequality and from a quantitative slow-variation hypothesis. The approach developed herein is illustrated through several examples in which the discussed theory is applied not only to DDEs, but also to semidiscrete parabolic problems given by partial delay differential equations with a diffusion term.