Semiparametric models encompass a huge family of models that have a fully parametric (finite number of parameters) with a nonparametric part. In general, the parametric part will be linear, and the semiparametric part will be treated as nuisance; but this is not always the case. One example where a semiparametric model would be relevant, could be for example modeling the ice-cream sales in terms of the weather and the price. It's likely that the sales-weather relationship is highly nonlinear (sales are really high when the temperature is high, but low when the temperature is moderate), whereas the price-sales one could be quite linear. In that case, we would want to treat the price effect as linear and the rest as nuisance.