Optical lattices have an important role in advancing our understanding of correlated quantum matter. The recent implementation of orbital degrees of freedom in chequerboard(1,2) and hexagonal(3) optical lattices opens up a new avenue towards discovering novel quantum states of matter that have no prior analogues in solid-state electronic materials. Here, we predict that an exotic topological semimetal emerges as a parity-protected gapless state in the orbital bands of a two-dimensional fermionic optical lattice. This new quantum state is characterized by a parabolic band-degeneracy point with Berry flux 2 pi, in sharp contrast to the pi flux of Dirac points as in graphene. We show that the appearance of this topological liquid is universal for all lattices with D-4 point-group symmetry, as long as orbitals with opposite parities hybridize strongly with each other and the band degeneracy is protected by odd parity. Turning on inter-particle repulsive interactions, the system undergoes a phase transition to a topological insulator whose experimental signature includes chiral gapless domain-wall modes, reminiscent of quantum Hall edge states.