  We address the quantum estimation of parameters encoded into the initial
state of two modes of a Dirac field described by relatively accelerated
parties. By using the quantum Fisher information (QFI), we investigate how the
weak measurements performed before and after the accelerating observer, affect
the optimal estimation of information encoded into the weight and phase
parameters of the initial state shared between the parties. Studying the QFI,
associated with weight parameter $ \vartheta $, we find that the acceleration
at which the optimal estimation occurs may be controlled by weak measurements.
Moreover, it is shown that the post-measurement plays the role of a quantum key
for manifestation of the Unruh effect. On the other hand, investigating the
phase estimation optimization and assuming that there is no control over the
initial state, we show that the weak measurements may be utilized to match the
optimal $ \vartheta $ to its predetermined value. Moreover, in addition to
determination of a lower bound on the QFI with the local quantum uncertainty,
we unveil an important upper bound on the precision of phase estimation in our
relativistic scenario, given by the maximal steered coherence (MSC). We also
obtain a compact expression of the MSC for general X states.
