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?
 A: If you're trying to work out how many df to use in your t or F distribution, you need to count the mean-parameters and not the variance parameter.
If you need to work out how many parameters you have in a BIC or an AICc say, then you should be counting the variance parameter. (On the other hand, for AIC itself, it won't matter whether you do or not, since it will just shift everything by the same constant. Unless you're doing something fairly unusual like comparing models with different numbers of variance parameters; in the general case, include all the parameters.)
A: 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).
