Lead sulfate is poorly soluble in water (about 30 mg per liter). Addition of HCl, especially a concnetrated HCl increases solubility of PbSO4.
There are two equilibrium processes that can cause this change
H3O+ + SO42- = H2O + HSO4-
Pb(HSO4)2 and PbCl2 are much better soluble in water than PbSO4 itself.
Moreover in a conc. HCl a formation of a complex anion PbCl3- should be taken into account, but the first reaction is much more important.