Using 281 $\textrm{pb}^{-1}$ of data collected with the CLEO-c detector at the $\psi(3770)$ resonance, we measured branching fractions of the Cabibbo-suppressed decays $D^0 \to \pi^+ \pi^-$, $D^0 \to \pi^0 \pi^0$, and $D^+ \to \pi^+ \pi^0$. We used these branching fraction measurements to extract the ratio of isospin amplitudes $|A_2 / A_0| = 0.429\pm0.018\pm0.017$ and the relative phase $\delta = (85.5 \pm 3.5 \pm 4.1)^{\circ}$ between these amplitudes, where the uncertainties are statistical and systematic, respectively. Our findings are consistent with, and lend support to the recently published official CLEO-c results.