First, the means and covariance of the assumed gaussian distributions in each class need to be estimated. When $p=1$, which means only one predictor, that is to estimate the class specific mean $\mu_k$ and the common variance $\sigma^2$. They can be estimated as follows $\hat{\mu}_{k}=\frac{1}{n_{k}} \sum_{i: y_{i}=k} x_{i}$ $\hat{\sigma}^{2}=\frac{1}{n-K} \sum_{k=1}^{K} \sum_{i: y_{i}=k}\left(x_{i}-\hat{\mu}_{k}\right)^{2}$ where $n$ is the total number of training observations, and $n_k$ is the number of training observations in the kth class. When $p>1$, which means multiple predictors, the estimation would be similar but much more complicated with $p(p+1)/2$ parameters in the covariance matrix.

As for the prior probabilty $\pi_k$, it can be estimated based on the proportion of class $k$ observations in the training set, which would be