Solved – Through an example: what is parametric and non-parametric boostrap

I'll give you an example. Let's say you have the data set $x_1,dots,x_n$, which you think comes from the normal distribution.

In parametric bootstrapping, you estimate the parameters of normal distribution $hatmu,hatsigma$, then you generate new sample from $x_1^*,dots,x^*_nsimmathcal{N}(hatmu,hatsigma^2)$

You can generate as many samples $x_1^*,dots,x^*_n$ as needed for you Monte Carlo simulation.

In non-parametric bootstrapping, you build empirical distribution function (EDF), then generate the sample $x_1^*,dots,x^*_n$ directly from EDF, not from the estimated normal distribution as in parametric bootstrapping.

It happens so that in some applications non-parametric bootstrapping leads to biased estimation, while parametric is unbiased, e.g. see G. Jogesh Babu, Eric D. Feigelson, "Astrostatistics: Goodness-of-Fit and All That!", in Astronomical Data Analysis Software and Systems XV, ASP Conference Series, Vol. 351, 2006. The paper is about K-S test critical values estimation.