  We give a complete characterization of those $f: [0,1] \to X$ (where $X$ is a
Banach space which admits an equivalent Fr\'echet smooth norm) which allow an
equivalent $C^2$ parametrization. For $X=\R$, a characterization is well-known.
However, even in the case $X=\R^2$, several quite new ideas are needed.
Moreover, the very close case of parametrizations with a bounded second
derivative is solved.
