  Low sonic Mach number shocks form in the intracluster medium (ICM) during the
formation of the large-scale structure of the universe. Nonthermal cosmic-ray
(CR) protons are expected to be accelerated via diffusive shock acceleration
(DSA) in those ICM shocks, although observational evidence for the $\gamma$-ray
emission of hadronic origin from galaxy clusters has yet to be established.
Considering the results obtained from recent plasma simulations, we improve the
analytic test-particle DSA model for weak quasi-parallel ($Q_\parallel$)
shocks, previously suggested by \citet{kang2010}. In the model CR spectrum, the
transition from the postshock thermal to CR populations occurs at the injection
momentum, $p_{\rm inj}$, above which protons can undergo the full DSA process.
As the shock energy is transferred to CR protons, the postshock gas temperature
should decrease accordingly and the subshock strength weakens due to the
dynamical feed of the CR pressure to the shock structure. This results in the
reduction of the injection fraction, although the postshock CR pressure
approaches an asymptotic value when the CR spectrum extends to the relativistic
regime. Our new DSA model self-consistently accounts for such behaviors and
adopts better estimations for $p_{\rm inj}$. With our model DSA spectrum, the
CR acceleration efficiency ranges $\eta\sim10^{-3}-0.01$ for supercritical,
$Q_\parallel$-shocks with sonic Mach number $2.25\lesssim M_{\rm s}\lesssim5$
in the ICM. Based on \citet{ha2018b}, on the other hand, we argue that proton
acceleration would be negligible in subcritical shocks with $M_{\rm s}<2.25$.
