Consider a symmetric Markovian jump process $\{X_t\}$ on a metric measure space $(M, d, μ)$. Chen, Kumagai, and Wang recently showed that two-sided heat kernel estimates and the parabolic Harnack inequality are both stable under bounded perturbations of the jumping measure, assuming $(M, d, μ)$ satisfies the volume-doubling and reverse-volume-doubling conditions. These results do not apply if $(M, d, μ)$ is a graph (or more generally, if $M$ contains any atoms $x$ such that $μ(x)>0$) because it is impossible for reverse-volume-doubling to hold on a space with atoms. We generalize the results of Chen, Kumagai, and Wang to a larger class of metric measure spaces, including all infinite graphs with volume-doubling. Our main tool is the construction of an "auxiliary space" that smooths out the atoms. We show that many properties transfer from $(M, d, μ)$ to the auxiliary space, and vice versa, including heat kernel estimates, the parabolic Harnack inequality, and their stable characterizations.