every set intersection for every set in a family with another family of sets

Outer[Intersection, p1, p2, 1]

{{{6}}, {{6}}, {{3, 5, 6}}}

From mobile, so not tested

Intersection@@#& /@ Tuples[{p1, p2}]

Update. Another solution, which preserves the inner/outer structure:

Table[Intersection[l1, l2], {l1, p1}, {l2, p2}]