  We investigate the ultimate precision achievable in Gaussian quantum
metrology. We derive general analytical expressions for the quantum Fisher
information matrix and for the measurement compatibility condition, ensuring
asymptotic saturability of the quantum Cram\'er-Rao bound, for the estimation
of multiple parameters encoded in multimode Gaussian states. We then apply our
results to the joint estimation of a phase shift and two parameters
characterizing Gaussian phase covariant noise in optical interferometry. In
such a scheme, we show that two-mode displaced squeezed input probes with
optimally tuned squeezing and displacement fulfil the measurement compatibility
condition and enable the simultaneous estimation of all three parameters, with
an advantage over individual estimation schemes that quickly rises with
increasing mean energy of the probes.
