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. Comment: 12 pages, 8 figures plus Appendix; comments are welcome
