  Quantum computation relies on accurate measurements of qubits not only for
reading the output of the calculation, but also to perform error correction.
Most proposed scalable silicon architectures utilize Pauli blockade of triplet
states for spin-to-charge conversion. In recent experiments, there have been
instances when instead of conventional triplet blockade readout, Pauli blockade
is sustained only between parallel spin configurations, with $|T_0\rangle$
relaxing quickly to the singlet state and leaving $|T_+\rangle$ and
$|T_-\rangle$ states blockaded -- which we call \textit{parity readout}. Both
types of blockade can be used for readout in quantum computing, but it is
crucial to maximize the fidelity and understand in which regime the system
operates. We devise and perform an experiment in which the crossover between
parity and singlet-triplet readout can be identified by investigating the
underlying physics of the $|T_0\rangle$ relaxation rate. This rate is tunable
over four orders of magnitude by controlling the Zeeman energy difference
between the dots induced by spin-orbit coupling, which in turn depends on the
direction of the applied magnetic field. We suggest a theoretical model
incorporating charge noise and relaxation effects that explains quantitatively
our results. Investigating the model both analytically and numerically, we
identify strategies to obtain on-demand either singlet-triplet or parity
readout consistently across large arrays of dots. We also discuss how parity
readout can be used to perform full two-qubit state tomography and its impact
on quantum error detection schemes in large-scale silicon quantum computers.
