What is the number of parameters for AIC if some coefficients are zero? I am estimating a linear regression model. I have n independent variables. One of the estimated coefficients is exactly zero. Does this coefficient count as a parameter when calculating the AIC?
 A: [The first two paragraphs address an earlier version of the OP.]
For AIC comparisons, you cannot change the dependent variable without accounting for it. So you cannot directly compare the AIC of a model for the original variable with the AIC of a model for the z-score of the original variable. 
Also, you are not quite right when you say I could arbitrarily increase the AIC by including further zero-coefficients for fictional variables without efficiently changing my model. These coefficients are fixed as opposed to being estimated, so they do not add extra penalty to AIC. The penalty is intended to account for overfitting, which involves estimation. If you are not estimating but rather imposing coefficient values, you do not overfit, so there is no need to penalize.

Does an estimated coefficient of zero counts as a parameter for AIC calculation?

I do not have a definite answer. Intuitively, I think it should, as the potential to overfit was there, although it did not materialize. But what are the chances of getting a coefficient value exactly equal to zero?
On the other hand, for example, in LASSO models an unbiased estimator of the degrees of freedom is the number of nonzero coefficients. So if the coefficient ended up being estimated at zero, it does not add up in the degrees of freedom calculation. So probably the answer depends on the model at hand?
