  We discuss the ultimate precision bounds on the multiparameter estimation of
single- and two-mode pure Gaussian states. By leveraging on previous approaches
that focused on the estimation of a complex displacement only, we derive the
Holevo Cram\'er-Rao bound (HCRB) for both displacement and squeezing parameter
characterizing single and two-mode squeezed states. In the single-mode
scenario, we obtain an analytical bound and find that it degrades monotonically
as the squeezing increases. Furthermore, we prove that heterodyne detection is
nearly optimal in the large squeezing limit, but in general the optimal
measurement must include non-Gaussian resources. On the other hand, in the
two-mode setting, the HCRB improves as the squeezing parameter grows and we
show that it can be attained using double-homodyne detection.
