  Non-detection of $\gamma$-ray emission from galaxy clusters has challenged
diffusive shock acceleration (DSA) of cosmic-ray (CR) protons at weak
collisionless shocks that are expected to form in the intracluster medium. As
an effort to address this problem, we here explore possible roles of Alfv\'en
waves self-excited via resonant streaming instability during the CR
acceleration at parallel shocks. The mean drift of Alfv\'en waves may either
increase or decrease the scattering center compression ratio, depending on the
postshock cross-helicity, leading to either flatter or steeper CR spectra. We
first examine such effects at planar shocks, based on the transport of Alfv\'en
waves in the small amplitude limit. For the shock parameters relevant to
cluster shocks, Alfv\'enic drift flattens the CR spectrum slightly, resulting
in a small increase of the CR acceleration efficiency, $\eta$. We then consider
two additional, physically motivated cases: (1) postshock waves are isotropized
via MHD and plasma processes across the shock transition and (2) postshock
waves contain only forward waves propagating along with the flow due to a
possible gradient of CR pressure behind the shock. In these cases, Alfv\'enic
drift could reduce $\eta$ by as much as a factor of 5 for weak cluster shocks.
For the canonical parameters adopted here, we suggest $\eta\sim10^{-4}-10^{-2}$
for shocks with sonic Mach number $M_{\rm s}\approx2-3$. The possible reduction
of $\eta$ may help ease the tension between non-detection of $\gamma$-rays from
galaxy clusters and DSA predictions.
