In this paper we consider a nonlinear nonlocal boundary value problem for a system of hyperbolic equations. By introducing new unknown functions, the nonlinear nonlocal boundary value problem for a system of hyperbolic equations is reduced to an equivalent boundary value problem for the integral-differential partial differential equation.  A boundary value problem containing a family of Cauchy problems for a system of integral-differential Fredholm equations with a parameter and an unknown function is investigated using the method of introducing additional functional parameters. A modified algorithm of Jumabayev's parameterization method of finding the solution of the family of boundary value problems is proposed. Application of the parameterization method leads to a system of non-linear implicit Fredholm-type integral equations with respect to parameters. Iterative methods are used to solve this system. Sufficient conditions for the existence of an isolated solution of the considered nonlinear nonlocal boundary value problem for the system of hyperbolic equations.