This paper addresses the random vibrations of the oscillators with correlated external and parametric excitations being Gaussian white noises. The exponential polynomial closure method is used in the analysis, with which the probability density of the system responses is obtained. Two oscillators are analyzed. One is about the linear oscillator subjected to correlated external and parametric excitations. Another is about the oscillator with cubic nonlinearity and subjected to correlated external and parametric excitations. Numerical studies show that exponential polynomial closure method provides computationally efficient and relatively accurate estimates of the stationary probabilistic solutions, particularly in the tail regions of the probability density functions.Numerical results further show that correlated external and parametric excitations can cause unsymmetrical probabilistic solutions and nonzero means which are different from those when the external and parametric excitations are independent.