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In this paper, a Parallel Direct Eigensolver for Sequences of Hermitian Eigenvalue Problems with no tridiagonalization is proposed, denoted by \texttt{PDESHEP}, and it combines direct methods with iterative methods. \texttt{PDESHEP} first reduces a Hermitian matrix to its banded form, then applies a spectrum slicing algorithm to the banded matrix, and finally computes the eigenvectors of the original matrix via backtransform. Therefore, compared with conventional direct eigensolvers, \texttt{PDESHEP} avoids tridiagonalization, which consists of many memory-bounded operations. In this work, the iterative method in \texttt{PDESHEP} is based on the contour integral method implemented in FEAST. The combination of direct methods with iterative methods for banded matrices requires some efficient data redistribution algorithms both from 2D to 1D and from 1D to 2D data structures. Hence, some two-step data redistribution algorithms are proposed, which can be $10\times$ faster than ScaLAPACK routine \texttt{PXGEMR2D}. For the symmetric self-consistent field (SCF) eigenvalue problems, \texttt{PDESHEP} can be on average $1.25\times$ faster than the state-of-the-art direct solver in ELPA when using $4096$ processes. Numerical results are obtained for dense Hermitian matrices from real applications and large real sparse matrices from the SuiteSparse collection.