Conformality of a map f is invariant under parametric transform

Question

Let S, S* be parametric surfaces given by parametrizations $\overrightarrow{r}$: U $\rightarrow$ S and $\overrightarrow{r*}$: U* $\rightarrow$ S*. Is the conformality of a map f : S $\rightarrow$ S* invariant under a proper parametric transform ($\phi(\hat{u},\hat{v})$,$\psi(\hat{u},\hat{v})$) of $\overrightarrow{r}$ from $\hat{U}$ to U? Prove or disprove?