Differential equations with involution give rise to nonlocal constraints that significantly limit the application of classical methods in the theory of boundary value problems. In this paper, we study a boundary value problem for a system of differential equations with involution. By applying the parameterization method, the original boundary value problem is reduced to an equivalent Cauchy problem and a system of linear algebraic equations with respect to the introduced parameters. Explicit analytical solvability conditions are obtained, and the spectral properties of the problem are investigated. In cases where the solvability conditions are not satisfied, the spectrum of the corresponding boundary value problem is analyzed. The obtained results extend the existing analytical approaches for studying boundary value problems for differential equations with involution.