In a GLM, we don't usually use the T-distribution unless we use a quasilikelihood. See relevant question here.
Let's assume that's what you mean; a GLM with quasilikelihood (variance proportional to some function of mean). An ordinary linear regression is the most common case. The distribution that the test-statistics take under the null is the same for each parameter when ideal modeling assumptions are met: T with $n-p$ degrees of freedom ($p$ number of parameters in model).
Thus it's trivial to report this DF more than once (for each parameter as you say). Furthermore, if the reader knows how many parameters are in the model, it's trivial to report the DF at all; nevertheless, some choose to report it anyway.
Most published studies will at least let you know the $n$, and if it matters, they're explicit about the $p$, but even when not, the $p$-value and confidence interval are calculated from this T-distribution. At the very least, this assures the reader that appropriate control has been made for the nuissance parameter of unknown residual variance.