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.
Parametric simply means that a set of parameters govern the nature of the (joint) probability distribution of assets, some of those parameters being the correlations. It is not true in general to state that a parametric VaR model has cross-correlation of assets as zero.
I have never used a model that specifically precludes correlations. But if you defined one as such then your equations would be reduced as you state, but it is a very stringent assumption.