  Let $G$ be a simple graph on $n$ vertices and $\mathcal{I}_G$ denotes parity
binomial edge ideal of $G$ in the polynomial ring $S = \mathbb{K}[x_1,\ldots,
x_n, y_1, \ldots, y_n].$ We obtain a lower bound for the regularity of parity
binomial edge ideals of graphs. We then classify all graphs whose parity
binomial edge ideals have regularity $3$. We classify graphs whose parity
binomial edge ideals have pure resolution.
