Traditional pervious concrete poses significant challenges in optimizing both mechanical properties and permeability. To address this issue, a novel type of self-compacting recycled pervious concrete (SCRPC) featuring vertical and penetrating channels has been developed. The vertical channels were created by pulling out the reinforcement in the pre-drilled holes that were artificially created in the mold, after the concrete had been poured. However, whether this concrete has superior durability and can be employed in different sulfate drying-wetting situations remains to be investigated. This study explored the sulfate resistance and permeability of SCRPC under five drying-wetting exposure regimes: full soaking in Na2SO4 solution with drying-wetting ratios of 3:18, 9:12, and 18:3; semi-soaking in Na2SO4 solution; and full soaking in MgSO4 solution. The results showed that the SCRPC soaked in MgSO4 solution suffered the largest compressive strength loss (13.4%) after 150 drying-wetting cycles. Furthermore, as the drying-wetting ratio increased, the sulfate degradation of the SCRPC increased. Despite the comparable relative dynamic modulus of elasticity of SCRPC after full soaking (95.54%) and semi-soaking (92.89%), ettringite and gypsum were identified as the predominant sulfate deterioration products of SCRPC, respectively. In contrast to the two stages for traditional pervious concrete, the effective porosity of SCRPC was divided into three stages during sulfate attack: an initial rapid decline stage, a subsequent increase stage, and a final slow decline stage. The permeability coefficient of SCRPC varied from 6.00 to 6.82 mm/s under different sulfate drying-wetting exposures. In summary, SCRPC has superior sulfate resistance and permeability, and it could be more applicable in environments containing Na2SO4 compared to MgSO4. This study provides basic data for the enhancement and application of pervious concrete with artificial vertical and penetrating channels.