Magnetic islands resulting from magnetic reconnection play significantly important roles in critical issues of magnetic fusion plasmas, because they essentially change magnetic topology and significantly degrade plasma confinement. In this paper, we report that the wide islands can lead to a structure bifurcation of flow parity inside islands, then induce a state with a large vortex flow. The mechanism is identified as an instability in the configuration with magnetic islands, in which the eigenmode parity is contrary to the conventional tearing mode and the radial profile of zonal flow also changes. The critical values of the width of magnetic island for the bifurcation are obtained and discussed. The dependence of bifurcation on various physical parameters is studied in detail. It is found that tearing instability parameter and diamagnetic drift flow have significant impacts on the bifurcation. Moreover, the characteristics of the bifurcation show the qualitative agreement with the experimental observation on the Large Helical Device.