= Solution
The <semicubical parabola> has <coordinate ring>
$$
A=k[x,y]/(x^2-y^3)\cong k[t^3,t^2]\subset k[t],
$$
under $x\mapsto t^3$, $y\mapsto t^2$. The isomorphism is valid in every characteristic: after division by the monic <polynomial> in $x$, a remainder $P(y)+xQ(y)$ maps to a sum of distinct even powers and odd powers at least three, and can vanish only when both <polynomials> are zero.
Its <fraction field> contains $t=t^3/t^2$, which is an <integral element> over $A$ because it satisfies $z^2-y=0$. However $t\notin A$, since a <polynomial> in $t^2,t^3$ has no term of degree one. Thus $A$ is not an <integrally closed domain>. The <ring> $k[t]$ of the affine line is a <principal ideal domain> and therefore an <integrally closed domain>. This property is preserved by <ring isomorphism>. By coordinate-ring functoriality,
$$
\boxed{Z(x^2-y^3)\not\cong\mathbb A^1}.
$$
The parametrization is a bijection on points over an algebraically closed <field>, but that does not make its inverse a morphism at the cusp.
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