Unitals can be obtained as closures of affine unitals via parallelisms. The isomorphism type of the closure depends on the chosen parallelism, which need not be unique. For affine $\operatorname{SL}(2,q)$-unitals, we introduce a class of parallelisms for odd order and one for square order. Translations are automorphisms of unitals, fixing each block through a given center. For each of the known parallelisms of affine $\operatorname{SL}(2,q)$-unitals, we compute all possible translations with centers on the block at infinity.