A related lesson, when one does perform a parametric analysis, is that simple and interpretable models are best. They certainly don't know anything about unbalanced designs, residual analysis, simultaneous inference, or any sort of non-normal analysis including categorical analysis. I have found it necessary to teach many statistical methodologies, especially logistic regression and log-linear analysis, to students at the same time as introducing data sets to which they are applicable. Categorical, especially dichotomous, data is so common that it seems indefensible to send students out as statistics graduates who know nothing about its analysis. Having settled on a parametric model, one will further test its assumptions by examining the residuals.
Residual analysis is now very well established, and students accept it very easily.
The conclusions of an analysis must be presented in the concrete terms of the physical problem, not in the terminology of mathematical statistics. The conclusions will be supported by plots or tables, either of the parameters of the formal model or, more often, the corresponding data display. The presentation of conclusions emphasizes Fisher's principle of data summary. Data summary is the basic process of data analysis, i.e. summarizing what the data has to say about the questions of interest. Tables and descriptive statistics summarize data in a obvious way.