I'm using NonlinearModelFit to fit 5 parameters to some data points and the "ParameterConfidenceIntervalTable" gives me the estimates with the CIs. Now I want to know if the estimated values of the parameters are statistically different.

Obviously this is the case if the CIs don't overlap, but unfortunately this can also be the case if there is a slight overlap (see here). So the question is:

What is the proper way to test if the parameter estimates are significantly different?