Using notation from Hartshone's Algebraic Geometry (see below), the projective closure of $\mathcal Z(y-x^2) \subset \mathbb A^2$ in $\mathbb P^2$ is the closure (in $\mathbb P^2$) of $$\begin{align*} \mathcal Z(xz-y^2) \cap U_x &= \{[a:b:c] \mid ac=b^2, a \ne 0\}\\ &=\{[a:b:c] \mid c=b^2/a, a \ne 0\}\\ &=\{[1:b/a:b^2/a^2] \mid a \ne 0\}. \end{align*}$$ where $U_x$ the complement of the zero set of $x$ in $\mathbb P^2$.
How do we proceed in computing the projective closure?
Relevant information from Hartshone's Algebraic Geometry:
To compute the defining ideal of the projective closure, you have to compute the homogenization of the defining ideal of the affine variety $I=(y-x^2)$. In general, to compute the homogenization of an ideal, you have to compute a Groebner basis of it, then homogenize all elements in the Groebner basis. Here, since $I$ is principle, its homogenization is just $I^h=(yz-x^2)$. Hence, the projective closure is $\mathcal{Z}(yz-x^2)$.