  The admixture of spin-singlet and spin-triplet pairing states in
superconductors can be typically induced by breaking spatial inversion
symmetry. Employing the {\it numerically exact} auxiliary-field Quantum Monte
Carlo method, we study such mixed-parity pairing phenomena of attractive
fermions with Rashba spin-orbit coupling (SOC) in two-dimensional optical
lattice at finite temperature. We systematically demystify the evolution of the
essential pairing structure in both singlet and triplet channels versus the
temperature, fermion filling, SOC and interaction strengths, via computing the
condensate fraction and pair wave function. Our numerical results reveal that
the singlet channel dominates in the fermion pairing and the triplet pairing
has relatively small contribution to the superfluidity for physically relevant
parameters. In contrast to the singlet channel mainly consisted of the on-site
Cooper pairs, the triplet pairing has plentiful patterns in real space with the
largest contributions from several nearest neighbors. As the SOC strengh
increases, the pairing correlation is firstly enhanced and then suppressed for
triplet pairing while it's simply weakened in singlet channel. We have also
obtained the Berezinskii-Kosterlitz-Thouless transition temperatures through
the finite-size analysis of condensate fraction. Our results can serve as
quantitative guide for future optical lattice experiments as well as accurate
benchmarks for theories and other numerical methods.
