We consider a family of nonlinear boundary value problems for integro-differential equations. An algorithm for finding its solutions based on the Dzhumabaev parameterization method is proposed. Conditions for the feasibility and convergence of these algorithms are obtained. The same conditions are sufficient conditions for the existence of an isolated solution in some ball to a family of nonlinear boundary value problems for integro-differential equations. When solving nonlinear problems, iterative processes are often used, so the choice of the initial approximation is an important part of the process of finding a solution. In order to demonstrate the application of a family of nonlinear boundary value problems for integro-differential equations, a test example of determining the initial approximation of a nonlinear nonlocal boundary value problem for a system of two hyperbolic equations with mixed derivatives is given by reducing this problem to a family of nonlinear boundary value problems for integro-differential equations and finding its initial approximations.