We introduce a parity–sum statistic on permutations that assigns to each position a weight determined by the parity of the entry occupying it. When restricted to alternating permutations, this statistic yields two q–analogues of the Euler numbers, corresponding to the up–down and down–up types. We establish symmetry and reciprocity properties of these polynomials. Specializing at q = 1, the resulting recursions reduce to the classical enumerative relations and recover André’s convolution identity for the Euler numbers. The distribution of the parity–sum statistic over the full symmetric group is also determined.