  We extend the definition of tridendriform bialgebra by introducing a weight
q. The subspace of primitive elements of a q-tridendriform bialgebra is
equipped with an associative product and a natural structure of brace algebra,
related by a distributive law. This data is called q-Gerstenhaber-Voronov
algebras. We prove the equivalence between the categories of connected
q-tridendriform bialgebras and of q-Gerstenhaber-Voronov algebras. The space
spanned by surjective maps, as well as the space spanned by parking functions,
have natural structures of q-tridendriform bialgebras, denoted ST(q) and
PQSym(q)*, in such a way that ST(q) is a sub-tridendriform bialgebra of
PQSym(q)*. Finally we show that the bialgebra of M-permutations defined by T.
Lam and P. Pylyavskyy may be endowed with a natural structure of
q-tridendriform algebra which is a quotient of ST(q).
