= Solution
The log density is
$$
-\log(2\sigma)-\frac{|y-\mu|}{\sigma}.
$$
Its kink occurs at the unknown location $\mu$, so its parameter dependence cannot be separated into a fixed finite-dimensional statistic of $y$ and the smooth canonical form required of an exponential dispersion family. Thus the two-parameter location-scale <Laplace distribution> is not such a family.
For independent observations, the <log-likelihood> is
$$
\ell(\beta,\sigma)
=-n\log(2\sigma)-\frac1\sigma
\sum_{i=1}^n|Y_i-X_i^T\beta|
=-n\log(2\sigma)-\frac{S(\beta)}{\sigma}.
$$
For every fixed $\sigma>0$, maximizing this expression in $\beta$ is equivalent to minimizing $S(\beta)$. Hence
$$
\boxed{
\widehat\beta\in\operatorname*{arg\,min}_\beta
\sum_{i=1}^n|Y_i-X_i^T\beta|
},
$$
the <least absolute deviations> estimator. At this value,
$$
\frac{\partial\ell}{\partial\sigma}
=-\frac n\sigma+\frac{S(\widehat\beta)}{\sigma^2},
$$
so, in the nondegenerate case $S(\widehat\beta)>0$,
$$
\boxed{\widehat\sigma=\frac{S(\widehat\beta)}n}.
$$
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