Comparison of symbolic and ordinary powers of parity binomial edge ideals

In this paper, we investigate when symbolic and ordinary powers of the parity binomial edge ideal of a graph fail to be equal. It turns out that if ${\mathcal{I}}_G$ is the parity binomial edge ideal of a graph $G$, then in each of the following cases the symbolic power ${\mathcal{I}}_G^{(t)}$ and the ordinary power ${\mathcal{I}}_G^t$ are not equal for some $t$: (i) the clique number of $G$ is greater than 3; (ii) $G$ has a net; or (iii) $G$ has a PT as an induced subgraph.