  We obtain explicit $p$-Wasserstein distance error bounds between the
distribution of the multi-parameter MLE and the multivariate normal
distribution. Our general bounds are given for possibly high-dimensional,
independent and identically distributed random vectors. Our general bounds are
of the optimal $\mathcal{O}(n^{-1/2})$ order. Explicit numerical constants are
given when $p\in(1,2]$, and in the case $p>2$ the bounds are explicit up to a
constant factor that only depends on $p$. We apply our general bounds to derive
Wasserstein distance error bounds for the multivariate normal approximation of
the MLE in several settings; these being single-parameter exponential families,
the normal distribution under canonical parametrisation, and the multivariate
normal distribution under non-canonical parametrisation. In addition, we
provide upper bounds with respect to the bounded Wasserstein distance when the
MLE is implicitly defined.
