I am going to examine the effectiveness of an intervention study. My advisor wants me to use parametric tests when testing the effectiveness of the intervention. Based on the central limit theory, I need to assign more than 30 people to intervention and control groups to be able to use parametric tests. In these circumstances, I must lead at least three intervention groups. Can I take 10 people instead of 30 people in the experimental group and bootstrap to analyze the data using parametric tests?
I'm not 100% clear but it seems that you may have misunderstood the central limit theorem. There is no specific value of n which ensures the central limit holds (in those cases where it applies). Rate at which the sampling distribution of the mean converges on the normal distribution depends on the skew and kurtosis and shape of the distribution being sampled. If it is unimodal, with mild skew and without heavy tails it may be fine to use parametric tests even with relatively small n. Likewise the the sampling distribution of the mean can be very non-normal even with huge n if skew is severe, the distribution multimodal and with heavy tails.
Bootstrapping can be helpful in these cases, but it isn't great with very small n. One useful check can be to compare results from parametric approaches with more robust approaches such as rank-transformation tests (Mann-Witney or Wilcoxon for instance). If there is a big discrepancy it suggests that the violation of assumptions is quite severe and the more conservative/robust procedure should be preferred.
Finally the distribution assumptions apply to the residuals of your model - so its usually best to look at the residual diagnostics rather than the raw data.
Ceren Bektaş-Aydın, the idea that bootstrap is the "magic" solution for inference in small samples is a sort of urban legend right now... and for some reason that I never really understood... I have seen moderately educated people arguing that small sample problems can be "solved" with bootstrap... well, it is not the case. Bootstrap is a nice solution when you do not know the distribution of an estimator, but it requires moderately to large samples to work well in general. The sample size that you need can be calculated in advance depending on the effect size, for a given power, that you want to detect, given your specific model. Let's say anyway that you end up with a small sample, or that the phenomenon itself it is a small sample one, then there are specific techniques to analyze these small samples, for example Bayesian methods, which work well in small samples as well as in large samples. See for example:
Article Modeling Hyperinflation Phenomenon: A Bayesian Approach
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