  We determine the retarded and advanced Green's functions and Hadamard
parametrices in curved spacetimes for linearized massive and massless gauge
bosons and linearized Einstein gravity with a cosmological constant in general
linear covariant gauges. These vector and tensor parametrices contain
additional singular terms compared with their Feynman/de Donder-gauge
counterpart. We also give explicit recursion relations for the Hadamard
coefficients, and indicate their generalization to $n$ dimensions. Furthermore,
we express the divergence and trace of the vector and tensor Green's functions
in terms of derivatives of scalar and vector Green's functions, and show how
these relations appear as Ward identities in the free quantum theory.
