  In this paper, we investigate some parity factors by using Lov\'asz's
(g,f)-parity theorem. Let $m>0$ be an integer. Firstly, we obtain a sufficient
and necessary condition for some graphs to have a parity factor with restricted
minimum degree. Secondly, we obtain some sufficient conditions for a graph to
have a parity factor with minimum degree $m$ in term of edge connectivity.
