Ionut, here is a link from a dissertation concerning multiple mediation; however, I am having difficulty finding power analysis. I am attaching a link for an article that indicates various sample sizes and suggest bootstrapping as an alternative.
You can simulate matrix in R or R studio, and then to form the serial mediation model. You can then check the fit indicators (e. g. chi-square/df) for the simulated model with exact number of participants (cases), and compare for which cases (e. g. 100, 200, 500, 1000 participants or cases) chi-square/df is significanlly lower.