  We unveil a topological phase of interacting fermions on a two-leg ladder of
unequal parity orbitals, derived from the experimentally realized double-well
lattices by dimension reduction. $Z_2$ topological invariant originates simply
from the staggered phases of $sp$-orbital quantum tunneling, requiring none of
the previously known mechanisms such as spin-orbit coupling or artificial gauge
field. Another unique feature is that upon crossing over to two dimensions with
coupled ladders, the edge modes from each ladder form a parity-protected flat
band at zero energy, opening the route to strongly correlated states controlled
by interactions. Experimental signatures are found in density correlations and
phase transitions to trivial band and Mott insulators.
