  This work completes the classification of the cubic vertices for arbitrary
spin massless bosons in three dimensions started in a previous companion paper
by constructing parity-odd vertices. Similarly to the parity-even case, there
is a unique parity-odd vertex for any given triple $s_1\geq s_2\geq s_3\geq 2$
of massless bosons if the triangle inequalities are satisfied ($s_1<s_2+s_3$)
and none otherwise. These vertices involve two (three) derivatives for odd
(even) values of the sum $s_1+s_2+s_3$. A non-trivial relation between
parity-even and parity-odd vertices is found. Similarly to the parity-even
case, the scalar and Maxwell matter can couple to higher spins through current
couplings with higher derivatives. We comment on possible lessons for 2d CFT.
We also derive both parity-even and parity-odd vertices with Chern-Simons
fields and comment on the analogous classification in two dimensions.
