This link (and others, e.g. slides 43 and 46 of this) say that:
Where all the coefficients in the model are point estimates, we could calculate the MSE to generate distributions for the distribution of future forecast estimates. Note that if we assume that the residuals in the model are, $\varepsilon_t \sim$ i. i. d. $\mathcal{N}\left(0,\sigma_{\varepsilon}^2\right)$, it implies that the forecast errors should also be normally distributed, $y_{t+h}-\dot{y}_t(h) \sim\mathcal{N}\left(0, \sigma_t(h)\right) $ Such that, $\frac{y_{t+h}-\dot{y}_t(h)}{\sqrt{\dot{\sigma}_t(h)}} \sim \mathcal{N}(0,1). $ In this case we could make use of a normal distribution, with $z_\alpha$ defining the upper and lower bounds that may be used to derive the forecast interval for the $h$-period ahead forecast, $ \left[\dot{y}_t(h)-z_{\alpha / 2} \sqrt{\dot{\sigma}_t(h)}, \quad \dot{y}_t(h)+z_{\alpha / 2} \sqrt{\dot{\sigma}_t(h)}\right]. $ This allows for the construction of standard confidence intervals for the parameter estimates, where the predictor, $\dot{y}_t(h)$, and the MSE, $\sigma_t(h)$, are used to derive the appropriate interval.
However, these forecast intervals estimates don't seem to take into account uncertainty in the fitted parameters. Why is this so?