  It is shown that for a $N$-boson system the parity of $N$ can be responsible
for a qualitative difference in the system response to variation of a
parameter. The nonlinear boson model is considered, which describes tunneling
of boson pairs between two distinct modes $X_{1,2}$ of the same energy and
applies to a Bose-Einstein condensate in an optical lattice. By varying the
lattice depth one induces the parity-dependent quantum switching, i.e. $X_1\to
X_2$ for even $N$ and $X_1\to X_1$ for odd $N$, for arbitrarily large $N$. A
simple scheme is proposed for observation of the parity effect on the
\textit{mesoscopic scale} by using the bounce switching regime, which is
insensitive to the initial state preparation (as long as only one of the two
$X_l$ modes is significantly populated), stable under small perturbations and
requires an experimentally accessible coherence time.
