A Markovian queueing model is considered in which servers of various types work in parallel to process jobs from a number of classes at rates mu(ij) that depend on the class, i, and the type, j. The problem of dynamic resource allocation so as to minimize a risk-sensitive criterion is studied in a law-of-large-numbers scaling. Letting X-i(t) denote the number of class-i jobs in the system at time t, the cost is given by E exp{n[integral(T)(0) h((X) over bar (t))dt + g((X) over bar (T))]}, where T > 0, h and g are given functions satisfying regularity and growth conditions, and (X) over bar = (X) over bar (n) = n(-1) X(n.). It is well known in an analogous context of controlled diffusion, and has been shown for some classes of stochastic networks, that the limit behavior, as n -> infinity, is governed by a differential game (DG) in which the state dynamics is given by a fluid equation for the formal limit phi of (X) over bar, while the cost consists of integral(T)(0) h(phi(t))dt + g(phi(T)) and an additional term that originates from the underlying large-deviation rate function. We prove that a DG of this type indeed governs the asymptotic behavior, that the game has value, and that the value can be characterized by the corresponding Hamilton-Jacobi-Isaacs equation. The framework allows for both fixed and a growing number of servers N -> infinity, provided N = o(n).