Variational inference use ELBO as the objective function, which is an expectation over some space. Computation of expectation is difficult because it is a high dimensional integral. The parameterization trick as usually used in PyMC3 and mainstream package relied on using samples to compute the expectation; and using expectation of the gradient to substitute computing the gradient of the expectation. You can find some good reference in https://arxiv.org/pdf/1610.02287.pdf (see equation 1 and 3), and a more recent paper https://arxiv.org/pdf/1805.08498.pdf (also see eq 1 and 3)