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Bayesian inference for exponential family random graph models (ERGMs) is a doubly-intractable problem because of the intractability of both the likelihood and posterior normalizing factor. Auxiliary variable based Markov Chain Monte Carlo (MCMC) methods for this problem are asymptotically exact but computationally demanding, and are difficult to extend to modified ERGM families. In this work, we propose a kernel-based approximate Bayesian computation algorithm for fitting ERGMs. By employing an adaptive importance sampling technique, we greatly improve the efficiency of the sampling step. Though approximate, our easily parallelizable approach is yields comparable accuracy to state-of-the-art methods with substantial improvements in compute time on multi-core hardware. Our approach also flexibly accommodates both algorithmic enhancements (including improved learning algorithms for estimating conditional expectations) and extensions to non-standard cases such as inference from non-sufficient statistics. We demonstrate the performance of this approach on two well-known network data sets, comparing its accuracy and efficiency with results obtained using the approximate exchange algorithm. Our tests show a wallclock time advantage of up to 50% with five cores, and the ability to fit models in 1/5th the time at 30 cores; further speed enhancements are possible when more cores are available.