Given a graph with colored edges, a Hamiltonian cycle is called alternating if its successive edges differ in color. The problem of finding such a cycle, even for 2-edge-colored graphs, is trivially NP-complete, while it is known to be polynomial for 2-edge-colored complete graphs. In this paper we study the parallel complexity of finding such a cycle, if any, in 2-edge-colored complete graphs. We give a new characterization for such a graph admitting an alternating Hamiltonian cycle which allows us to derive a parallel algorithm for the problem. Our parallel solution uses a perfect matching algorithm putting the alternating Hamiltonian cycle problem to the RNC class. In addition, a sequential version of our parallel algorithm improves the computation time of the fastest known sequential algorithm for the alternating Hamiltonian cycle problem by a factor of $O(\sqrt {n} )$.