  $\operatorname{SL}(2,q)$-unitals are unitals of order $q$ admitting a regular
action of $\operatorname{SL}(2,q)$ on the complement of some block. They can be
obtained from affine $\operatorname{SL}(2,q)$-unitals via parallelisms. We
compute a sharp upper bound for automorphism groups of affine
$\operatorname{SL}(2,q)$-unitals and show that exactly two parallelisms are
fixed by all automorphisms. In $\operatorname{SL}(2,q)$-unitals obtained as
closures of affine $\operatorname{SL}(2,q)$-unitals via those two parallelisms,
we show that there is one block fixed under the full automorphism group.
