  In this paper, we studied the $B_{(s)} \to (D_{(s)},\bar{D}_{(s)}) \rho \to
(D_{(s)}, \bar{D}_{(s)})\pi \pi$ decays by employing a framework for the
quasi-two-body decays in the perturbative QCD (PQCD) factorization approach. We
use the two-pion distribution amplitudes $\Phi_{\pi\pi}$, which contains both
resonant and nonresonant contributions from the pion pair, to describe the
final state interactions (FSIs) between the pions in the resonant region. We
found that (a) for all considered decays, the PQCD predictions for their
branching ratios based on the quasi-two-body and the two-body framework agree
well with each other due to ${\cal B}(\rho \to \pi\pi) \approx 100\%$; For
$B^+\to \bar{D^0}\rho^+\to \bar{D^0}\pi^+ \pi^0$ and other four considered
decay modes, the PQCD predictions do agree well with the measured values within
errors; (b) the great difference between the PQCD predictions for ${\cal
B}(B\to \bar{D} \rho \to \bar{D} \pi\pi)$ and ${\cal B}(B\to D \rho \to D
\pi\pi)$ can be understood by the strong CKM suppression factor $ R_{\rm CKM}
\approx 3 \times 10^{-4}$; (c) for the $B_s\to D \rho \to D \pi\pi$ and $B_s\to
\bar{D} \rho \to \bar{D} \pi\pi$ decays, however, the PQCD predictions of
$R_{\rm s1}\approx 0.13$ and $R_{\rm s2}\approx 0.14$ do agree very well with
the moderate CKM suppression factor $R^s_{\rm CKM}\approx 0.14$; and (d) the
PQCD predictions for the ratios $R_{D\rho}$ and the strong phase difference
$\cos\delta_{D\rho}$ of the three $B \to \bar{D} \rho$ decay modes agree well
with the LHCb measurements within one standard deviation.
