Sorry, but I think that to do a correlation you don't neccesarily need replicates. If for example you want to know if there is a correlation between temperature in summer and development of algal blooms you don' t have replicates and still you will be able to do it.

Correct, however, to compare between a control line and a treatment line, you do need replicates, so that you can establish if the difference(s) seen are due to natural variation or are actually statistically valid difference.

Either parametric or non-parametric, I think I know the difference. Parametric tests such anova when you have normal and homocedastic data, non parametric, such as anova on ranks (kruskal-wallis) when any of those test fail, am I wrong?

Nearly but not quite, Parametric tests are indeed for distributed data (assuming the data fits a distribution is the primary criterion), but they also require large (n>30) sample sizes, and assume certain things about your data, meaning that they are not particularly robust tests, despite their power. Non-parametric are used where one or more of those conditions are not met, and can still be used on data that fits a distribution - they have less power, but are more robust.

It is incredibly common to see the wrong tests used in the literature, I'm not sure if this is because of lack of statistical knowledge (to be honest I know bugger all about stats myself) or if it is due to poor reviewing of the papers. I suspect, because parametric tests are much easier to calculate, this is the reason they are taught primarily in schools and universities, despite not being particularly useful in the real world.