Systems of parameters consisting of linear forms for monomial ideal quotients

Let $S=K[x_1,\dots ,x_n]$ be a polynomial ring over a field $K$ and let $I$ be a monomial ideal of $S$. We classify linear systems of parameters of $S\mathrm{/}I$ over any field $K$ using linear algebra and show explicit linear systems of parameters when $K$ has at least $n$ elements. If $I(G)$ is the edge ideal of a perfect graph $G$, a cycle or the complement of a cycle, we show that $S\mathrm{/}I(G)$ has a 0-1 linear system of parameters, and for graphs with independence number equal to $2$, we characterize when $S\mathrm{/}I(G)$ has a 0-1 linear system of parameters.