Linear in the parameters means that you can write your prediction as
$$\beta_0+\sum_{j=1}^px_{ij}\beta_j $$
For some definition of $x_{ij} $. But these x's need not be linear functions of your data. For example, ploynomial fitting of a time series has $x_{ij}=t_i^j $ where $t_i $ is the time associated with data point $i $. The prediction is a non linear function of time, but it is linear in the betas.
UPDATE
In response to the comment, the answer is "sort of". If $\beta_2$ was constant, then the predictor is linear in $\beta_0,\beta_1,\exp (\beta_3) $. It is not linear in $\beta_3$, but a transformation of $\beta_3$. In terms of least squares estimates it doesn't make much difference here.