Let $\mathbb{A}_{2}$ denote the 2-affice space over $\mathbb{C}$. Now let $Y=V(y-x^{2})$, here $V$ denotes the zero-locus. Now by definition the coordinate ring is the quotient ring $k[x,y]/I(Y)$ where $I(Y)$ is the ideal generated by all polynomials vanishing on $Y$.
Now the author computes $k[x,y]/I(Y)$, then it computes $k[x,y]/