Difference-in-differences (DID) analysis is used widely to estimate the causal effects of health policies and interventions. A critical assumption in DID is "parallel trends": that pre-intervention trends in outcomes are the same between treated and comparison groups. To date, little guidance has been available to researchers who wish to use DID when the parallel trends assumption is violated. Using a Monte Carlo simulation experiment, we tested the performance of several estimators (standard DID; DID with propensity score matching; single-group interrupted time-series analysis; and multi-group interrupted time-series analysis) when the parallel trends assumption is violated. Using nationwide data from US hospitals (n = 3737) for seven data periods (four pre-interventions and three post-interventions), we used alternative estimators to evaluate the effect of a placebo intervention on common outcomes in health policy (clinical process quality and 30-day risk-standardized mortality for acute myocardial infarction, heart failure, and pneumonia). Estimator performance was assessed using mean-squared error and estimator coverage. We found that mean-squared error values were considerably lower for the DID estimator with matching than for the standard DID or interrupted time-series analysis models. The DID estimator with matching also had superior performance for estimator coverage. Our findings were robust across all outcomes evaluated.