Why does barium sulphate not dissolve in HCl while barium sulphite does?
$\ce{BaSO_3}$ has the same property as $\ce{BaCO_3}$ :
When reacting with an acidic solution, these two substances produce a gas which can get out of the solution, and displaces the equilibrium to the right  : $$\ce{BaCO_3 + 2 H^+ -&gt;  Ba^{2+} + CO_2(g) + H_2O}$$
$$\ce{BaSO_3 + 2 H^+ -&gt; Ba^{2+} + H_2O + SO_2(g)}$$ This allows these two substances to be soluble in $\ce{HCl}$ solutions. As a contrary $\ce{BaSO_4}$ has not this possibility to produce a gas in an acidic solution : it is and remains insoluble in $\ce{HCl}$ solutions.