Hybridization is a reconfiguring/mixing of atomic orbitals that occurs to produce maximum bonding overlap. This is important in sigma bond formation. But with pi bonds, the atomic orbitals already happen to be "optimally" oriented for maximum overlap and therefore do not gain from hybridization.
Mathematically, p orbitals have + and - phases above and below the plane. When mixed with S orbitals (+ phase), the opposite phase lobe gets almost wiped out while the other lobe gets enhanced. That's good for sigma bonding, but not good for pi bonding which needs both ends above and below the plane.