Based on the Central Limit Theorem, would it still be appropriate to use parametric tests (e.g., independent t-tests, ANOVA, or ANCOVA) despite this non-normality? If the sample size is about 100 (50 for the control group and 50 for the experimental group).
When the sample size is that great you can use parametric tests according to the education in Finland.
It depends.
1) Note that the assumption isn't that the data are normal. You can look a plot of residuals from the analysis to see how far from normal the residuals are.
2) If the type of variable is really non-normal, you may want to use a model that is more appropriate for that kind of data. For example, some things in nature are log-normally distributed. It's usually not particularly valuable to test the means on this kind of variable. Here, someone might log transform the variable to compare geometric means or use Gamma regression.
2b) The bottom line is that you want to do an analysis that is meaningful for your data and what you want to know.
3) Note that the CLT is an asymptotic result. You don't know a priori what sample size is needed for the distribution of the means to be approximately normal.
The Central Limit Theorem tells you that the sampling distribution will approach normality if the sample is large. Most books and empirical studies show that large means 30 or more degrees of freedom.
As Sal Mangiafico has already said, it depends (on how non-normal the data are and in what way they are non-normal. Not trying to put words in your mouth, Sal, so please say so if I have misunderstood you.)
This is a perennial question in statistics, and statisticians seem to be divided into two camps on it. One camp says that if the data set is "large enough," then it is OK to use parametric tests on non-normal data (but opinions vary on the definition of "large enough"). The other camp says no, if the data are sufficiently non-normal, you should use non-parametric tests (opinions also vary on what is "sufficiently non-normal" and on the best way/s to test normality). And as previously mentioned, the way/s in which the data are non-normal also matter in this decision. I am in the second camp (although, for full disclosure, I do not have formal qualifications in statistics).
To illustrate just how misleading parametric tests can be on non-parametric data, I wrote a short tutorial paper with some examples last time I saw a similar question on Researchgate:
Preprint A Cautionary Note on Statistical Testing
I can't give a definite answer on your data set, Cecilia, without seeing it, but I suggest that you try both parametric testing and non-parametric testing and see what answers you get. If they give you similar answers (they generally won't be identical), then parametric testing is OK for your data set. If they give you significantly different answers, then trust the non-parametric test result rather than the parametric test result. You may need to take into account that non-parametric tests tend to have lower statistical efficiency than parametric tests - meaning that they need a bigger data set to achieve the same confidence level.
I guess this is a roundabout way of saying that you can (almost) always trust a non-parametric test, so if you are unsure about whether you can trust a parametric test with the data you have, then just use the non-parametric test to avoid sleepless nights.