Subsequence Search

I don't know how extensible this is, but it works for your case,

Cases[list,
    p[args:PatternSequence[___, a, _p, q, ___]] :> SequenceCases[
        {args},
        {a, p[x_, y_], q} :> {x, y}
    ],
    Infinity
]
(* {{{1, 2}, {3, 4}, {5, 6}}} *)

This also works for your list, but like Jason B, I don't that is will be applicable to your XML problem.

Cases[Partition[List @@ list[[1]], 3, 1], {a, u : p[_, _], q} -> u]

{p[1, 2], p[3, 4], p[5, 6]}