Exactly how many parameters are in multiple linear regression? (some sources include sigma squared, some don't) I've read some resources such as my Introductory Econometrics textbook that state the model parameters are $B_{0}$, $B_{1}$, ..., $B_{p}$ where $p$ is the number of predictors in the model.
Other resources include $\sigma^{2}$ as well as one of the 'model parameters'.
Is there a reason for the discrepancy?

The regression standard error $\sigma$ is an unknown parameter, so it ought really be included when mentioning the parameters of the regression model.  Nevertheless, in some contexts the focus is on the parameters $B_0,...,B_p$ that enter into the regression function; there are a number of regression results that are affected by the parameters in the regression function where these are the focus.  Consequently, some authors speak a bit loosely when that is their focus and they exclude consideration of $\sigma$ (though they probably shouldn't).
