In this paper, we study a partial information optimal control problem derived by forward-backward stochastic systems with correlated noises between the system and the observation. Utilizing a direct method, an approximation method, and a Malliavin derivative method, we establish three versions of maximum principle (i.e., necessary condition) for optimal control. To show their applications, we work out two illustrative examples within the frameworks of linear-quadratic control and recursive utility and then solve them via the maximum principles and stochastic filtering.