  We study a sensor network of distributed Mach-Zehnder interferometers (MZIs)
for the parallel (simultaneous) estimation of an arbitrary number $d \geq 1$ of
phase shifts. The scheme uses a squeezed-vacuum state that is split between $d$
modes by a quantum circuit (QC). Each output mode of the QC is the input of one
of $d$ MZIs, the other input of each MZI being a coherent state. In particular,
${\it i}$) we identify the optimal configuration of the sensor network that
allows the estimation of any linear combination of the $d$ phases with maximal
sensitivity. The protocol overcomes the shot-noise limit and reaches Heisenberg
scalings with respect to the total average number of particles in the overall
probe state, the multiphase estimation only requiring local photocounting.
Furthermore, the parallel multiphase estimation overcomes optimal separable
strategies for the estimation of any linear combination of the phases: the
sensitivity gain being a factor $d$, at most. Viceversa, ${\it ii}$) given a
specific QC, we identify the optimal linear combination of the phases that
maximizes the sensitivity and show that results are robust against random
choices of the QC. Our scheme paves the ways to a variety of applications in
distributed quantum sensing.
