Parametric VaR assumption question

Why do you have to make a correlation matrix when calculating the parametric value at risk, if one of the assumptions for this method to work is that the assets of the portfolio must be independently distributed (i.e. their correlation must be equal 0)? Furthermore, $Var(X+Y) = Var(X) + Var(Y) + 2\cdot Cov(X,Y)$ is used when expanding the variance to get the portfolio variance, but this can only be used also if $X$ and $Y$ are not independent.