  We propose a physical scheme for the realization of two-dimensional
topological odd-parity superfluidity in a spin-independent bond-centered square
optical lattice based upon interband fermion pairing. The D4 point-group
symmetry of the lattice protects a quadratic band crossing, which allows one to
prepare a Fermi surface of spin-up fermions with odd parity close to the
degeneracy point. In the presence of spin-down fermions with even parity
populating a different energetically well separated band, odd-parity pairing is
favored. Strikingly, as a necessary prerequisite for pairing both Fermi
surfaces can be tuned to match well. As a result, topological superfluid phases
emerge in the presence of merely s-wave interaction. Due to the Z2 symmetry of
these odd-parity superfluids, we infer their topological features simply from
the symmetry and the Fermi-surface topology as confirmed numerically.
