There are at least two kinds of uncertainty associated with parameter estimates when statistical models are fit to sample data. The first kind of uncertainty is typically conveyed by confidence regions which provide a plausible range of values for population parameters of interest. The second kind of uncertainty involves a sensitivity analysis (Cook, 1986) with respect to model fit. Here, contours representing alternative solutions for parameter estimates that are almost as good as the optimal estimates in terms of model fit are obtained. Contours of these slightly suboptimal parameter values have been termed fungible weights or contours (Waller, 2008; Waller & Jones, 2009; MacCallum, Browne & Lee, 2009). Although distinct from each other, confidence regions and fungible contours communicate parameter uncertainty and are both computed from the likelihood function. Given these commonalities, we set out to clarify the relationship between confidence regions and fungible parameter contours by accomplishing three objectives. First, we show that confidence regions and fungible parameter contours are analytically related when both types of parameter uncertainty are unified under a general perturbation framework. Second, we carried out a simulation study that confirms the distinction between confidence regions and fungible parameter contours. Although the magnitude of correlations among measured variables have an impact on these two kinds of parameter uncertainty, confidence regions are primarily determined by sample size while fungible parameter contours are determined primarily by model fit and, to a much smaller extent, sampling variability. Third, we implemented a new computational procedure for obtaining confidence regions and fungible parameter contours associated with focal parameters by the profile likelihood method, which takes account of nuisance parameters. We conclude with directions for future research and end with a discussion of what applied researchers may gain from examining these two distinct kinds of parameter uncertainty.