yeah, that's why I had to use the parens saying that it was clearly data that would affect their stats (think about studies where you can only extract data from 5 rats per group, etc.). Also why I said normality-assumed parametric tests. There are many other parametric tests which aren't based on "normality" assumptions. But yes as you said, you can absolutely use normality-assumed parametric tests on non-normal data within reason and your conditions you listed are within reason. Of course this is my opinion and my opinion could just as easily be entirely wrong as decided by the expert community :D

And as always, your allowances depend on the test you are performing :P Also gotta love statistics for keeping a large list of customary exceptions determined by the community too.